one

Bahamian Creole: wan

Belizean Creole: wan

Bislama: wan

Gullah: one

Jamaican Creole: wan

Nigerian Pidgin: wọ́n, wọn

Sranan Tongo: wan

Tok Pisin: wan

→ Atong (India): wan

→ Fanagalo: wan

PIE word
 *h₁óynos
From Middle English oon, on, oan, an, from Old English ān (“one”), from Proto-West Germanic *ain, from Proto-Germanic *ainaz (“one”), from Proto-Indo-European *h₁óynos (“single, one”).
Cognate with Scots ae, ane, wan, yin (“one”); North Frisian ån (“one”); Saterland Frisian aan (“one”); West Frisian ien (“one”); Dutch een, één (“one”); German Low German een; German ein, eins (“one”); Danish en (“one”); Swedish en (“one”); Norwegian Nynorsk ein (“one”), Icelandic einn (“one”); Latin ūnus (“one”) (Old Latin oinos); Russian оди́н (odín), Spanish uno. Doublet of a, an, and Uno.
The use as an indefinite personal pronoun may have been influenced by unrelated French on, although the Germanic languages widely use cognates for the same sense (usually in non-subject function, but also in subject function, e.g. Luxembourgish een).
Verb form from Middle English onen.

inh

see for numerical forms in other scripts

one's damned if one does and one's damned if one doesn't

(number theory) The first positive number in the set of natural numbers.


prime number

composite number

antonym(s) of “natural number divisible only by itself and 1”

The fundamental theorem of arithmetic states that every natural number greater than 1 can be factorized into prime numbers in a way that is unique up to the order in which the factors are written.

2007, James Alfred Walker, Julian Francis Miller, Predicting Prime Numbers Using Cartesian Genetic Programming, Marc Ebner, Michael O'Neill, Anikó Ekárt, Anna Isabel Esparcia-Alcázar, Leonardo Vanneschi (editors), Genetic Programming: 10th European Conference, EuroGP 2007, Proceedings, Springer, LNCS 4445, page 215,
As the evolved solution for the first 16 prime numbers was capable of accepting inputs up to 31, we decided to extend the experiment to see how the solution generalised on 15 previously unseen inputs (just as we did with the integer-based approach). From the 15 unseen inputs, 7 of the predicted 15 outputs were prime numbers, which is just below 50%, indicating that the solution had learned something about "primeness" or favoured prime numbers.

The most common definition of a prime number used in school seems to be 'an integer whose only factors are one and itself', which unfortunately leaves open the question of whether one is itself a prime number. Until the nineteenth century, most mathematicians regarded one as a prime number – Henri Lebesgue (1875–1941) is often said to be the last professional mathematician to call one prime – so it is a little unfair to regard learners as silly for thinking this today, or for questioning why we do not now regard one as a prime number – it is still a good question.

2010, Colin Foster, Resources for Teaching Mathematics: 14-16, Continuum International Publishing Group, page 168:

Some poems, echoing the purpose of early poetic treatises on scientific principles, attempt to elucidate the mathematical concepts that underlie prime numbers. Others play with primes’ cultural associations. Still others derive their structure from mathematical patterns involving primes.

2013 July-August, Sarah Glaz, “Ode to Prime Numbers”, in American Scientist, volume 101, number 4:

(number theory) Any natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

(obsolete, number theory) Any natural number (including 1) that is divisible only by itself and 1.

prime

natural number >1 divisible only by itself and 1

Eisenstein prime number without imaginary part

Wedderburn-Etherington number prime number


irrational

en-adj

irrational (comparative more irrational, superlative most irrational)

more irrational (comparative)
most irrational (superlative)

From Latin irratiōnālis, from ir- + ratiōnālis.

The number π is irrational.

(mathematics, arithmetic, number theory, not comparable) Of a real number, that cannot be written as the ratio of two integers.


abundant number

en-noun

abundant number (plural abundant numbers)

The factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30, and 1 + 2 + 3 + 5 + 6 + 10 + 15 = 42, which is greater than 30, so 30 is an abundant number.

It has been shown that the largest odd number which cannot be written as the sum of two abundant numbers is 20161.

1970, Geometric Transformations III, Random House, page 128:

1992, Stanley Rabinowitz (editor), Index to Mathematical Problems, 1980-1984, MathPro Press, page 185,
(a) Let k be fixed. Do there exist sequences of k consecutive abundant numbers?

We shall not be concerned with abundant numbers in this book, nevertheless it may be helpful to use this historical example as an illustration. We note the property that any multiple of an abundant number is abundant.

1996, Richard R. Hall, Sets of Multiples, Cambridge University Press, page xi:

(number theory) A number that is less than the sum of its proper divisors (all divisors except the number itself).


LCD

lowest common denominator
lowest common denominator#English

least common denominator
least common denominator#English

(number theory) Initialism of lowest common denominator / least common denominator.


friendly

en-adj

friendly (comparative friendlier or more friendly, superlative friendliest or most friendly)

friendlier (comparative)
more friendly (comparative)
friendliest (superlative)
most friendly (superlative)

From Middle English frendly, freendly, frendely, frendlich, from Old English frēondlīċ, from Proto-Germanic *frijōndlīkaz, equivalent to friend + -ly. Cognate with Saterland Frisian früntelk, fjuntelk (“friendly”), West Frisian freonlik (“friendly”), Dutch vriendelijk (“friendly”), German Low German fründelk, frünnelk (“friendly”), German freundlich (“friendly”).

friendly

friendly pairs

friendly n-tuples

(number theory) Being or relating to two or more natural numbers with a common abundancy.


perfect number

Calque of Latin numerus perfectus, with this use of perfectus itself a semantic loan from Ancient Greek τέλειος (téleios).

The factors of 6 are 1, 2, 3 and 6, and 1 + 2 + 3 = 6, so 6 is a perfect number.

The factors of 28 are 1, 2, 4, 7, 14 and 28, and 1 + 2 + 4 + 7 + 14 = 28, so 28 is a perfect number.

(number theory) A number that is the sum of all of its divisors except itself.

nombre perfecte

a number that is the sum of all of its divisors except itself


excessive number

en-noun

excessive number (plural excessive numbers)


deficient number

deficient number (plural deficient numbers)

The factors of 22 are 1, 2 and 11 and 22, and 1 + 2 + 11 = 14, which is less than 22, so 22 is a deficient number.

(number theory) A number that is greater than the sum of all of its divisors except itself.


defective number

defective number (plural defective numbers)


radical

Inherited from Middle English radical, from Latin rādīcālis (“of or pertaining to the root, having roots, radical”).

radicale

ring theory: ideal of elements of a ring sharing a certain given property

(number theory) The product of the distinct prime factors of a given positive integer.

radicale

number theory: product of the distinct prime factors of a given positive integer


congruence

en-noun

congruence (countable and uncountable, plural congruences)

/ˈkɒŋ.ɡɹʊ.əns/

/ˈkɔŋ.ɡɹʉ.əns/

From Middle English congruence, from Latin congruentia (“agreement”), from congruēns, present active participle of congruō (“meet together, agree”).

համաձայնություն

the quality of agreeing or corresponding

(mathematics, number theory) A relation between two numbers indicating they give the same remainder when divided by some given number.

kongruence

number theory, relation indicating two numbers give the same remainder when divided by some number


Euler's totient function

Named after Swiss mathematician Leonhard Euler (1707–1783).

named-after

(number theory) The function that calculates the total number of integers below a given integer that are coprime to it.


normal

en-adj

normal (comparative normaler or more normal, superlative normalest or most normal)

normaler (comparative)
more normal (comparative)
normalest (superlative)
most normal (superlative)

From Latin normālis (“made according to a carpenter's square; later: according to a rule”), from nōrma (“carpenter's square”), of uncertain origin; doublet of normale. The earliest meaning of the word in English was "perpendicular; forming a right angle" like something normālis (“made according to a carpenter's square”), but by Late Latin normālis had also come to mean "according to a rule", from which modern English senses of the word derive: in the 1800s, as people began to quantitatively study things like height, weight and blood pressure, the usual or most common values came to be called "normal", and by extension values regarded as healthy or desirable came to be called "normal" regardless of their usuality.

der

A number whose individual digits in a given base representation follow a uniform distribution is said to be simply normal.

A number that is normal for every base b ≥ 2 is said to be absolutely normal.

uniform distribution
uniform distribution

According to norms or rules or to a regular pattern.

(mathematics) Adhering to or being what is considered natural or regular in a particular field or context:

(number theory, of a real number) In whose representation in a given base b ≥ 2, for every positive integer n, the bⁿ possible strings of n digits follow a uniform distribution.


prime

composite

From Middle English prime, from Old French prime and its etymon, Latin prīmus (“first”), from earlier prīsmos < *prīsemos < Proto-Italic *priisemos, ultimately from Proto-Indo-European *per- (“beyond, before”). Doublet of primo and primus.
The noun sense "apostrophe-like symbol" originates from the fact that the symbol ′ was originally a superscript Roman numeral one.

3 is a prime.

mathematical structure
mathematical structure

(algebra, number theory) A prime element of a mathematical structure, particularly a prime number.

просто число

algebra: prime element of a mathematical structure


prim

ro-adj

prim m or n (feminine singular primă, masculine plural primi, feminine and neuter plural prime)

primă (feminine singular)
primi (masculine plural)
prime (feminine neuter plural)
prim (indefinite masculine neuter singular)
primă (feminine indefinite singular)
primi (indefinite masculine singular)
prime (feminine indefinite neuter plural)
primul (definite masculine neuter singular)
prima (definite feminine singular)
primii (definite masculine singular)
primele (definite feminine neuter plural)
prime (feminine indefinite singular)
primului (definite masculine neuter singular)
primei (definite feminine singular)
primelor (definite masculine singular)
primilor (definite feminine neuter plural)


elementary

en-adj

elementary (comparative more elementary, superlative most elementary)

more elementary (comparative)
most elementary (superlative)

From Middle English elementare, from Latin elementārius (“elementary”), from elementum (“one of the four elements of antiquity; fundamentals”) + -ārius (adjective-forming suffix). Cognate with French élémentaire. By surface analysis, element + -ary.

(mathematics, of an argument or proof) Straightforward, employing only basic techniques; not requiring substantial knowledge (of some particular domain, object, etc.).

(number theory, of an argument or proof, mostly historical outside the phrase "Elementary number theory") Making no use of complex analysis.


prime factor

Hence, the prime factors of 100 are 1, 2, 2, 5, 5, and 1 × 2 × 2 × 5 × 5 = 100. From the above illustration, we derive the following rule for finding all the prime factors of any composite number.

1847, James Robinson, The American Arithmetic, John P. Jewett & Co., page 91:

Theorem 5.5 Dickman's Theorem. The probability of a number N chosen at random having a prime factor between N#92;alpha and N#123;#92;alpha(1#43;#92;delta)#125; is approximately #61;#92;delta, independent of the magnitude of #92;alpha, if #92;delta is small.

1994, Hans Riesel, Prime Numbers and Computer Methods for Factorization, Springer, page 161:

If any subset of the set of prime factors of a be taken, then the product of the elements of this subset is a factor of a.
For example, 12 is a factor of 60 and the set of prime factors of 12 is {2, 2, 3}, which is a subset of {2, 2, 3, 5}, which is the set of prime factors of 60.

2003, Gary R. Jensen, Arithmetic for Teachers: With Applications and Topics from Geometry, American Mathematical Society, page 180:

(number theory) A factor of a given integer which is also a prime number.


magic square

(number theory) An n-by-n arrangement of n² numbers such that the numbers in each row, in each column and along both diagonals all have the same sum.


transcendental

transcendental (comparative more transcendental, superlative most transcendental)

more transcendental (comparative)
most transcendental (superlative)

algebraic

antonym(s) of “not the root of a polynomial with rational coefficients”

algebraic

antonym(s) of “containing elements that are not the root of a polynomial”

LL-Q1860 (eng)-Flame, not lame-transcendental.wav

The theory of transcendental numbers was originated by Liouville in his famous memoir^† of 1844 in which he obtained, for the first time, a class, très-étendue, as it was described in the title of the paper, of numbers that satisfy no algebraic equation with integer coefficients.

1975, Alan Baker, Transcendental Number Theory, 2nd edition, Cambridge University Press, published 1990, page 1:

If the distribution of decimal digits of #92;pi (or any other transcendental number) is truly random (suspected but not yet mathematically proven!), given any arbitrary finite sequence of whole numbers, that sequence would be included an infinite number of times in the decimal expansion of #92;pi.

2005, Juan G. Roederer, Information and Its Role in Nature, Springer, page 28:

(algebra, number theory, field theory, of a number or an element of an extension field) Not algebraic (i.e., not the root of any polynomial that has positive degree and rational coefficients).


algebraic number

en-noun

algebraic number (plural algebraic numbers)

The golden ratio (φ) is an algebraic number since it is a solution of the quadratic equation x²+x-1=0, whose coefficients are integers.

The square root of a rational number, √, is an algebraic number since it is a solution of the quadratic equation nx²-m=0, whose coefficients are integers.

Thus, the equation x-eʸ#61;0 is satisfied for x#61;1,#92;y#61;0 and for no other pair of algebraic numbers.

1918, The American Mathematical Monthly, volume 25, Mathematical Association of America, page 435:

As a matter of fact, it is not true that the algebraic numbers above can be factored uniquely, but the first case of failure occurs when p = 23.

1921, L. J. Mordell, Three Lectures on Fermat's Last Theorem, page 16:

(i) It is easily checked that the set of all algebraic numbers is countable, whereas the set of all complex numbers is uncountable (this non-constructive proof goes back to Cantor).

1991, P. M. Cohn, Algebraic Numbers and Algebraic Functions, Chapman & Hall, page 83, The existence of such 'transcendental' numbers is well known and it can be proved at three levels

(algebra, number theory) A complex number (more generally, an element of a number field) that is a root of a polynomial whose coefficients are integers; equivalently, a complex number (or element of a number field) that is a root of a monic polynomial whose coefficients are rational numbers.

代數數 /代数数

element of a number field that is a root of a polynomial with integer coefficients


algebraic

en-adj

algebraic (comparative more algebraic, superlative most algebraic)

more algebraic (comparative)
most algebraic (superlative)

transcendental

antonym(s) of “that is the root of some polynomial”

transcendental

antonym(s) of “whose every element is the root of some polynomial”

/ˌæl.d͡ʒɪˈbɹeɪ.ɪk/

/ˌæl.d͡ʒɪˈbɹæɪ.ɪk/

Partly from algebra + -ic and partly from French algébraïque.

(algebra, number theory, of a number) Which is a root of some polynomial whose coefficients are rational.


adjoin

en-verb

adjoin (third-person singular simple present adjoins, present participle adjoining, simple past and past participle adjoined)

adjoins (present singular third-person)
adjoining (participle present)
adjoined (participle past)
adjoined (past)

From Anglo-Norman adjoindre, from Latin adiungō.

The ring adjunction #92;mathbb#123;Q#125;#91;#92;sqrt#123;2#125;#93; can be obtained from #92;mathbb#123;Q#125; by adjoining #92;sqrt#123;2#125; to #92;mathbb#123;Q#125;.

The field adjunction #92;mathbb#123;Q#125;(#92;pi) can be obtained from #92;mathbb#123;Q#125; by adjoining #92;pi to #92;mathbb#123;Q#125;.

(transitive, mathematics, chiefly algebra and number theory) To extend an algebraic object (e.g. a field, a ring, etc.) by adding to it an element not belonging to it and then ensuring closure, for example by also adding all finite power series of the element.

adjoindre

mathematics: to add (an element) to a structure


divisor

head

divisor c (singular definite divisoren, plural indefinite divisorer)

divisoren (definite singular)
divisorer (indefinite plural)
divisor (indefinite nominative singular)
divisoren (definite nominative singular)
divisorer (indefinite nominative plural)
divisorerne (definite nominative plural)
divisors (genitive indefinite singular)
divisorens (definite genitive singular)
divisorers (genitive indefinite plural)
divisorernes (definite genitive plural)


15≡3 mod 4, since both 15 and 3 gives a remainder of 3 when divided by 4.

(number theory) "Is congruent to"; used in calculations of modulo.


Legendre symbol

en-noun

Legendre symbol (plural Legendre symbols)

Named after French mathematician Adrien-Marie Legendre (1752–1833), who introduced the symbol in 1798 in his work Essai sur la Théorie des Nombres ("Essay on the Theory of Numbers").

1994, James K. Strayer, Elementary Number Theory, Waveland Press, 2002, Reissue, page 109,
Our only method at present for the computation of Legendre symbols requires a possible consideration of (p-1)/2 congruences (unless, of course, we are fortunate enough to encounter the desired quadratic residue along the way).

The Jacobi symbol, which generalizes the Legendre symbol, sheds some additional light on how to determine whether (7.29) has solutions when m has two or more distinct prime factors.

2006, Neville Robbins, Beginning Number Theory, 2nd edition, Jones & Bartlett, page 195:

Jacobi symbols can be used to facilitate the calculation of Legendre symbols. In fact, Legendre symbols can be eventually calculated by Jacobi symbols [17]. That is, the Legendre symbol can be calculated as if it were a Jacobi symbol. For example, consider the Legendre symbol #92;left(#92;frac#123;335#125;#123;2999#125;#92;right) where 335 = 5·67 is not a prime (of course, 2999 is a prime, otherwise, it is not a Legendre symbol).

2013, Song Y. Yan, Number Theory for Computing, Springer, page 149:

(number theory) A mathematical function of an integer and a prime number, written (a/p), which indicates whether a is a quadratic residue modulo p.


Jacobi symbol

Named after German mathematician Carl Gustav Jakob Jacobi, who introduced the notation in 1837.

2000, Song Y. Yan, Number Theory for Computing, Springer, 2000, Softcover reprint, page 114,
Although the Jacobi symbol ((1009)/(2307))=1, we still cannot determine whether or not the quadratic congruence 1009=x²(mod 2307) is soluble.
Remark 1.6.10. Jacobi symbols can be used to facilitate the calculation of Legendre symbols.

With more than two factors, having a Jacobi symbol of 1 only means that x may be a quadratic non-residue modulo an even number of factors only. Thus in the general case, the Jacobi symbol is not enough to test for the existence of a discrete logarithm. Thanks to this efficient test, given any public process, for example based on a hash function, that transforms the identity of a user into a number x modulo N, this number can directly be used as the user's public key if its Jacobi symbol is 1.

2009, Antoine Joux, “Chapter 1: Introduction to Identity-Based Cryptography”, in Marc Joye, Gregory Neven, editors, Identity-based Cryptography, IOS Press, page 8:

2014, Ibrahim Elashry, Yi Mu, Willy Susilo, Jhanwar-Barua's Identity-Based Encryption Revisited, Man Ho Au, Barbara Carminati, C.-C. Jay Kuo (editors), Network and System Security: 8th International Conference, Springer, LNCS 8792, page 279,
From the above equations, guessing the Jacobi symbol ((2y_is_j_1s_j_2+2)/N) from ((2y_j_1s_j_1+2)/N) and ((2y_j_2s_j_2+2)/N) is as hard as guessing them from independent Jacobi symbols.

(number theory) A mathematical function of integer a and odd positive integer b, generally written (a/b), based on, for each of the prime factors pᵢ of b, whether a is a quadratic residue or nonresidue modulo pᵢ.


quadratic reciprocity

The theorem highlights a particular form of reciprocity in the solvability of the quadratic equation a² = b in modular arithmetic. It was conjectured by Leonhard Euler and Adrien-Marie Legendre and first proved by Carl Friedrich Gauss.

Yes, but we need not only Quadratic Reciprocity but also Dirichlet's theorem on primes in arithmetic progressions to see this.

2007, Paul B. Garrett, Abstract Algebra, Taylor & Francis (Chapman Hall/CRC Press), page 287:

Gauss studied these sorts of numbers while attempting to formulate and prove higher order reciprocity laws, following his success with quadratic reciprocity.

2009, Sam Vandervelde, Circle in a Box, American Mathematical Society, page 153:

An interesting consequence of the above algorithm is that one can evaluate the Jacobi symbol in deterministic polynomial time in certain cases analogous to the way (“reduce and flip”) that one computes this symbol using quadratic reciprocity in the case F#61;#92;mathbb#123;Q#125;.

2013, J. Voight, “Identifying the Matrix Ring: Algorithms for Quaternion Algebras and Quadratic Forms”, in Krishnaswami Alladi, Manjul Bhargava, David Savitt, Pham Huu Tiep, editors, Quadratic and Higher Degree Forms, Springer, page 284:

(number theory) The mathematical theorem which states that, for given odd prime numbers p and q, the question of whether p is a square modulo q is equivalent to the question of whether q is a square modulo p.

Adrien-Marie Legendre
Carl Friedrich Gauss
Leonhard Euler
quadratic reciprocity


Pell's equation

Pell's equation (plural Pell's equations)

Named by Leonhard Euler after the 17th-century mathematician John Pell, whom Euler mistakenly believed to be the first to find a general solution.

However, due to Euler's mistake in attributing the equation to English mathematician John Pell (1610-1585), Equation (27.14) is called Pell's equation. Results concerning Pell's equation will be stated without proof.

1974, Allan M. Kirch, Elementary Number Theory: A Computer Approach, Intext Educational Publishers, page 212:

Thus (P,Q) satisfies Pell's equation and so by Lemma 1, P#47;Q is a convergent to #92;sqrtd.

1989, Mathematics Magazine, Volume 62, Mathematical Association of America, page 258:

We introduced Pell's equation
#92;qquad#92;qquad#92;qquad#92;qquadx²-ny²#61;1
in Chapter 4 as an example of a Diophantine equation. The solution x#61;577,#92;y#61;408 of the Pell equation x²-2y²#61;1 was used in India in the fourth century to produce the fraction #92;tfrac#123;577#125;#123;408#125; as an excellent rational approximation for #92;sqrt 2.
It is easy to see why solutions to Pell's equation can be used to approximate solutions to #92;sqrtn—this was known to Archimedes, who used this method to approximate square roots.

2013, John J. Watkins, Number Theory: A Historical Approach, Princeton University Press, page 409:

Diophantine equation
Diophantine equation

(number theory) The Diophantine equation x²-my²=1 for a given integer m, to be solved in integers x and y.


algebraic integer

algebraic integer (plural algebraic integers)

A Gaussian integer z=a+ib is an algebraic integer since it is a solution of either the equation z²+(-2a)z+(a²+b²)=0 or the equation z-a=0.

An algebraic number is said to be an algebraic integer if the coefficient of the highest power of x in the minimal polynomial P is 1. The algebraic integers in an algebraic number field k form a ring R.

1984, Alan Baker, A Concise Introduction to the Theory of Numbers, Cambridge University Press, page 62:

1989, Heinrich Rolletschek, Shortest Division Chains in Imaginary Quadratic Number Fields, Patrizia Gianni (editor), Symbolic and Algebraic Computation: International Symposium, Springer, LNCS 358, page 231,
Let O_d be the set of algebraic integers in an imaginary quadratic number field Q [√],d<0, where d is the discriminant of O_d.

We consider the problem of finding all algebraic integers which belong to a bounded subset of the complex plane together with their conjugates.

2010, Pierre Moussa, “Localisation of algebraic integers and polynomial iteration”, in Sergiy Kolyada, Yuri Nanin, Martin Möller, Pieter Moree, Thomas Ward, editors, Dynamical Numbers: Interplay Between Dynamical Systems and Number Theory, American Mathematical Society, page 83:

(algebra, number theory) A real or complex number (more generally, an element of a number field) which is a root of a monic polynomial whose coefficients are integers; equivalently, an algebraic number whose minimal polynomial (lowest-degree polynomial of which it is a root and whose leading coefficient is 1) has integer coefficients.

代數整數 /代数整数

algebraic number whose minimal polynomial has integer coefficients


Euclidean algorithm

Euclidean algorithm (plural Euclidean algorithms)

1985, Erich Kaltofen, Heinrich Rolletschek, Arithmetic in Quadratic Fields with Unique Factorization, Bob F. Caviness (editor), EUROCAL '85, European Conference on Computer Algebra, Linz, Proceedings, Volume 2, Springer, LNCS 204, page 279,
In a quadratic field Q (√), D a squarefree integer, with class number 1 any algebraic integer can be decomposed uniquely into primes but for only 21 domains Euclidean algorithms are known. We prove that for D<-19 even remainder sequences with possibly nondecreasing norms cannot determine the GCD of arbitrary inputs.

2003, Ali Akhavi, Brigitte Vallée, Average Bit-Complexity of Euclidean Algorithms, Ugo Montanari, Jose D.P. Rolim, Emo Welzl (editors), Automata, Languages and Programming: 27th International Colloquium, Proceedings, Springer, LNCS 1853, page 373,
In this paper, we provide new analyses that characterize the precise average bit-complexity of a class of Euclidean algorithms.
We consider here five algorithms that are all classical variations of the Euclidean algorithm and are called Classical (𝒢), By-Excess (ℒ), Centered (𝒦), Subtractive (𝒯) and Binary (ℬ).

The general behavior of lattice reduction algorithms is far from being well understood.[…]We explain how a mixed methodology has already proved fruitful for small dimensions p, corresponding to the variety of Euclidean algorithms (p = 1) and to the Gauss algorithm (p = 2).

2009, Brigitte Vallée, Antonio Vera, “3: Probabilistic Analyses of Lattice Reduction Algorithms”, in Phong Q. Nguyen, Brigitte Vallée, editors, The LLL Algorithm: Survey and Applications, Springer, page 71:

greatest common divisor
greatest common divisor

(arithmetic, number theory) Specifically, a method, based on a division algorithm, for finding the greatest common divisor (gcd) of two given integers; any of certain variations or generalisations of said method.


greatest common divisor

greatest common divisor (plural greatest common divisors)

least common multiple

antonym(s) of “largest number that divides each of a set of positive integers”

The greatest common divisor of 66, 30 and 18 is 6.

1974, John M. Peterson, Basic Concepts of Elementary Mathematics, Prindle, Weber & Schmidt, page 148,
Euclid's algorithm is a process for finding the greatest common divisor of any two whole numbers.

Suppose (a#95;0,a#95;1,#92;dots,a#95;n)#92;in#92;Z we say (a#95;0,a#95;1,#92;dots,a#95;n) is primitive if the greatest common divisor of a#95;0,a#95;1,#92;dots,a#95;n is 1.

2006, W. B. Vasantha Kandasamy, Florentin Smarandache, Neutrosophic Rings, Hexis, page 53:

This paper presents a regularization theory for numerical computation of polynomial greatest common divisors and a convergence analysis, along with a detailed description of a blackbox-type algorithm.[…]As one of the fundamental algebraic problems with a long history, finding the greatest common divisor (GCD) of univariate polynomials is an indispensable component of many algebraic computations besides being an important problem in its own right.

2011, Zhonggang Zeng, “The Numerical Greatest Common Divisor of Univariate Polynomials”, in Leonid Gurvits, Philippe Pébay, J. Maurice Rojas, David Thompson, editors, Randomization, Relaxation, and Complexity in Polynomial Equation Solving, American Mathematical Society, page 187:

(arithmetic, number theory) The largest positive integer (respectively polynomial, element of a given ring) that is a divisor of each of a given set of integers (respectively polynomials, elements of a given ring).


irreducible

en-adj

irreducible (comparative more irreducible, superlative most irreducible)

more irreducible (comparative)
most irreducible (superlative)

/ˌɪɹɪˈdjuːsɪbəl/

LL-Q1860 (eng)-Flame, not lame-irreducible.wav

(number theory, of a fraction) Whose numerator and denominator share no common factor greater than 1.

(number theory, of an integer) Unable to be factored into smaller integers; prime.

несъкратим

(mathematics, of an integer) unable to be factored into smaller integers


superabundant

LL-Q1860 (eng)-Flame, not lame-superabundant.wav

superabundant number
superabundant number

(number theory) Being or relating to a superabundant number.


Egyptian fraction

en-noun

Egyptian fraction (plural Egyptian fractions)

The fraction #92;frac#123;7#125;#123;10#125; can be written as the Egyptian fraction #92;frac#123;1#125;#123;2#125;#43;#92;frac#123;1#125;#123;5#125;.

The calculation of Mᴺ where both M and N are real numbers can be done by expressing N as the sum of an integer h and an Egyptian fraction 1#47;g#95;1#43;1#47;g#95;2#43;1#47;g#95;3#43;.... Then Mᴺ#61;Mʰ#92;sqrt#91;g#95;1#93;#123;M#125;#92;sqrt#91;g#95;2#93;#123;M#125;#92;sqrt#91;g#95;3#93;#123;M#125;... where each of the roots can be calculated by means of the Newton–Raphson method.

1995, C. Pomerance, A. Sárkőzy, Combinatorial number theory, R. L. Graham, M. Grötschel, L. Lovász (edtors), Handbook of Combinatorics, Elsevier (North-Holland), page 1014,
However, a representation of r as a sum of distinct Egyptian fractions is certainly not unique and this fact leads to many questions.

In [27], Erdős also considers various questions relating to Egyptian fraction decompositions of #92;textstyle 1#61;#92;sumⁿ#95;#123;k#61;1#125;#92;frac#123;1#125;#123;x#95;k#125;.

2013, R. L. Graham, “Paul Erdős and Egyptian Fractions”, in László Lovász, Imre Ruzsa, Vera T. Sós, editors, Erdős Centennial, Springer,, page 293:

The modern description of Egyptian fraction reckoning being "restricted" to unit fractions is obviously anachronistic (indeed, the Egyptian concept of fractions did not include a numerator, but from a historian's point of view this cannot be criticized on the basis that our modern fractions consist of denominator and numerator). Furthermore, this criticism does not do justice to the development of Egyptian fractions.

2016, Annette Imhausen, Mathematics in Ancient Egypt: A Contextual History, Princeton University Press, page 5:

(number theory) A representation of a rational number as a sum of distinct unit fractions.

egyptiläinen murtoluku

representation of a rational number as sum of distinct unit fractions


Diophantine equation

Diophantine equation (plural Diophantine equations)

1993, Ross Talent, Alf van der Poorten (translation editors), Vladimir G. Sprindžuk, Classical Diophantine Equations, Springer, Lecture Notes in Mathematics 1559, page viii,
Nevertheless, the fundamental idea of Skolem's method, the reduction of algebraic diophantine equations to exponential equations, has shown exceptional vitality in recent episodes of the theory of diophantine equations.

Based on lectures he^([Davenport]) gave at the University of Michigan in the early 1960s, this book is concerned with the use of analytic methods in the study of integer solutions to Diophantine equations and Diophantine inequalities.

2005, Harold Davenport, edited by T. D. Browning, Analytic Methods for Diophantine Equations and Diophantine Inequalities, 2nd edition, Cambridge University Press, page i:

2020, Hasan Sankari, Mohammad Abobala, Neutrosophic Linear Diophantine Equations With Two Variables, in Neutrosophic Sets and Systems, Volume 38, University of New Mexico, page 402,
The following theorem determines the criteria for the solvability of [a] neutrosophic linear Diophantine equation.

(number theory) A polynomial equation whose variables are only permitted to assume integer values.

Diophantine equation
simple:Diophantine equation

equació diofàntica

polynomial equation of integer variables


alkuluku

alkuluku (nominative singular)
alkuluvut (nominative plural)
alkuluku (accusative nominative singular)
alkuluvut (accusative nominative plural)
alkuluvun (accusative genitive singular)
alkuluvut (accusative genitive plural)
alkuluvun (genitive singular)
alkulukujen (genitive plural)
alkulukua (partitive singular)
alkulukuja (partitive plural)
alkuluvussa (inessive singular)
alkuluvuissa (inessive plural)
alkuluvusta (elative singular)
alkuluvuista (elative plural)
alkulukuun (illative singular)
alkulukuihin (illative plural)
alkuluvulla (adessive singular)
alkuluvuilla (adessive plural)
alkuluvulta (ablative singular)
alkuluvuilta (ablative plural)
alkuluvulle (allative singular)
alkuluvuille (allative plural)
alkulukuna (essive singular)
alkulukuina (essive plural)
alkuluvuksi (singular translative)
alkuluvuiksi (plural translative)
alkuluvutta (abessive singular)
alkuluvuitta (abessive plural)
alkuluvuin (instructive plural)
See the possessive forms below. (comitative)
alkulukuni (nominative singular)
alkulukuni (nominative plural)
alkulukuni (accusative nominative singular)
alkulukuni (accusative nominative plural)
alkulukuni (accusative genitive singular)
alkulukuni (accusative genitive plural)
alkulukuni (genitive singular)
alkulukujeni (genitive plural)
alkulukuani (partitive singular)
alkulukujani (partitive plural)
alkuluvussani (inessive singular)
alkuluvuissani (inessive plural)
alkuluvustani (elative singular)
alkuluvuistani (elative plural)
alkulukuuni (illative singular)
alkulukuihini (illative plural)
alkuluvullani (adessive singular)
alkuluvuillani (adessive plural)
alkuluvultani (ablative singular)
alkuluvuiltani (ablative plural)
alkuluvulleni (allative singular)
alkuluvuilleni (allative plural)
alkulukunani (essive singular)
alkulukuinani (essive plural)
alkuluvukseni (singular translative)
alkuluvuikseni (plural translative)
alkuluvuttani (abessive singular)
alkuluvuittani (abessive plural)
alkulukuineni (comitative plural)
alkulukusi (nominative singular)
alkulukusi (nominative plural)
alkulukusi (accusative nominative singular)
alkulukusi (accusative nominative plural)
alkulukusi (accusative genitive singular)
alkulukusi (accusative genitive plural)
alkulukusi (genitive singular)
alkulukujesi (genitive plural)
alkulukuasi (partitive singular)
alkulukujasi (partitive plural)
alkuluvussasi (inessive singular)
alkuluvuissasi (inessive plural)
alkuluvustasi (elative singular)
alkuluvuistasi (elative plural)
alkulukuusi (illative singular)
alkulukuihisi (illative plural)
alkuluvullasi (adessive singular)
alkuluvuillasi (adessive plural)
alkuluvultasi (ablative singular)
alkuluvuiltasi (ablative plural)
alkuluvullesi (allative singular)
alkuluvuillesi (allative plural)
alkulukunasi (essive singular)
alkulukuinasi (essive plural)
alkuluvuksesi (singular translative)
alkuluvuiksesi (plural translative)
alkuluvuttasi (abessive singular)
alkuluvuittasi (abessive plural)
alkulukuinesi (comitative plural)
alkulukumme (nominative singular)
alkulukumme (nominative plural)
alkulukumme (accusative nominative singular)
alkulukumme (accusative nominative plural)
alkulukumme (accusative genitive singular)
alkulukumme (accusative genitive plural)
alkulukumme (genitive singular)
alkulukujemme (genitive plural)
alkulukuamme (partitive singular)
alkulukujamme (partitive plural)
alkuluvussamme (inessive singular)
alkuluvuissamme (inessive plural)
alkuluvustamme (elative singular)
alkuluvuistamme (elative plural)
alkulukuumme (illative singular)
alkulukuihimme (illative plural)
alkuluvullamme (adessive singular)
alkuluvuillamme (adessive plural)
alkuluvultamme (ablative singular)
alkuluvuiltamme (ablative plural)
alkuluvullemme (allative singular)
alkuluvuillemme (allative plural)
alkulukunamme (essive singular)
alkulukuinamme (essive plural)
alkuluvuksemme (singular translative)
alkuluvuiksemme (plural translative)
alkuluvuttamme (abessive singular)
alkuluvuittamme (abessive plural)
alkulukuinemme (comitative plural)
alkulukunne (nominative singular)
alkulukunne (nominative plural)
alkulukunne (accusative nominative singular)
alkulukunne (accusative nominative plural)
alkulukunne (accusative genitive singular)
alkulukunne (accusative genitive plural)
alkulukunne (genitive singular)
alkulukujenne (genitive plural)
alkulukuanne (partitive singular)
alkulukujanne (partitive plural)
alkuluvussanne (inessive singular)
alkuluvuissanne (inessive plural)
alkuluvustanne (elative singular)
alkuluvuistanne (elative plural)
alkulukuunne (illative singular)
alkulukuihinne (illative plural)
alkuluvullanne (adessive singular)
alkuluvuillanne (adessive plural)
alkuluvultanne (ablative singular)
alkuluvuiltanne (ablative plural)
alkuluvullenne (allative singular)
alkuluvuillenne (allative plural)
alkulukunanne (essive singular)
alkulukuinanne (essive plural)
alkuluvuksenne (singular translative)
alkuluvuiksenne (plural translative)
alkuluvuttanne (abessive singular)
alkuluvuittanne (abessive plural)
alkulukuinenne (comitative plural)
alkulukunsa (nominative singular)
alkulukunsa (nominative plural)
alkulukunsa (accusative nominative singular)
alkulukunsa (accusative nominative plural)
alkulukunsa (accusative genitive singular)
alkulukunsa (accusative genitive plural)
alkulukunsa (genitive singular)
alkulukujensa (genitive plural)
alkulukuaan (partitive singular)
alkulukuansa (partitive singular)
alkulukujaan (partitive plural)
alkulukujansa (partitive plural)
alkuluvussaan (inessive singular)
alkuluvussansa (inessive singular)
alkuluvuissaan (inessive plural)
alkuluvuissansa (inessive plural)
alkuluvustaan (elative singular)
alkuluvustansa (elative singular)
alkuluvuistaan (elative plural)
alkuluvuistansa (elative plural)
alkulukuunsa (illative singular)
alkulukuihinsa (illative plural)
alkuluvullaan (adessive singular)
alkuluvullansa (adessive singular)
alkuluvuillaan (adessive plural)
alkuluvuillansa (adessive plural)
alkuluvultaan (ablative singular)
alkuluvultansa (ablative singular)
alkuluvuiltaan (ablative plural)
alkuluvuiltansa (ablative plural)
alkuluvulleen (allative singular)
alkuluvullensa (allative singular)
alkuluvuilleen (allative plural)
alkuluvuillensa (allative plural)
alkulukunaan (essive singular)
alkulukunansa (essive singular)
alkulukuinaan (essive plural)
alkulukuinansa (essive plural)
alkuluvukseen (singular translative)
alkuluvuksensa (singular translative)
alkuluvuikseen (plural translative)
alkuluvuiksensa (plural translative)
alkuluvuttaan (abessive singular)
alkuluvuttansa (abessive singular)
alkuluvuittaan (abessive plural)
alkuluvuittansa (abessive plural)
alkulukuineen (comitative plural)
alkulukuinensa (comitative plural)


continued fraction

continued fraction (plural continued fractions)

Several results focus on the famous Rogers–Ramanujan continued fraction [47], [48, pp. 214-215], the only continued fraction appearing in Ramanujan's published papers.

1992, G. E. Andrews, B. C. Berndt, L. Jacobsen, R. L. Lamphere, “The Continued Fractions Found in the Unorganized Portions of Ramanujan's Notebooks”, in Memoirs of the American Mathematical Society, volume 99, number 477, page 1:

2000, Andrew Zardecki, 18: Continued Fractions in Time Series Forec[a]sting, Da Ruan (editor), Fuzzy Systems and Soft Computing in Nuclear Engineering, Physica-Verlag, Studies in Fuzziness and Soft Computing, page 397,
We achieve this by using well-known examples from the number theory pertaining to the continued fractions. Any sequence of natural numbers drawn from the probability distribution of the quotients of the continued fraction corresponding to an irrational number represents a typical sequence, in the sense that almost all sequences of quotients have this distribution.

We have seen that all rational numbers, and no other number, can be expressed as finite simple continued fractions.
The main reason of interest of continued fractions, however, is in their application to the representation of irrational numbers. To that end we shall need infinite simple continued fractions.

2009, M. Welleda Baldoni, Ciro Ciliberto, G.M. Piacentini Cattaneo, translated by Daniele Gewurz, Elementary Number Theory, Cryptography and Codes, page 48:

(mathematics, number theory) A compound numerical expression consisting of an integer plus a fraction whose numerator is a positive integer and whose denominator is a continued fraction (an integer plus a fraction), and so on, with finite or infinite recursion.


prosti broj

sh-noun

prosti broj m (Cyrillic spelling прости број)

прости број (Cyrillic)


Riemann zeta function

en-noun

Riemann zeta function (usually uncountable, plural Riemann zeta functions)

Named after German mathematician Bernhard Riemann.

named-after

It is straightforward to show that the Riemann zeta function has zeros at the negative even integers and these are called the trivial zeros of the Riemann zeta function.

2004, Paula B. Cohen, “Interactions between number theory and operator algebras in the study of the Riemann zeta function (d'apres Bost-Connes and Connes)”, in David Chudnovsky, Gregory Chudnovsky, Melvyn Bernard Nathanson, editors, Number Theory: New York Seminar 2003, Springer,, page 87:

The Riemann zeta-function (which has no relation to the Weierstrass function of Chapter 8, and must not be confused with it) was originally of interest because of its connection with number theory. Since then it has served as the model for a proliferation of "zeta-functions" throughout mathematics. Some mention of the Riemann zeta-function, and treatment of the prime number theorem as an asymptotic result have become a topic treated by writers of introductory texts in complex variables.

2008, Sanford L. Segal, Nine Introductions in Complex Analysis, Revised edition, Elsevier (North-Holland), page 397:

2009, Arthur T. Benjamin, Ezra Brown (editors), Biscuits of Number Theory, Mathematical Association of America, page 195,
The Riemann zeta function is the function ζ(s)=∑ₙ₌₁ ᪲n⁻ˢ for s a complex number whose real part is greater than 1. […] The historical moments include Euler's proof that there are infinitely many primes, in which he proves
ζ(s)=∏ₚₚᵣᵢₘₑ(1-1/(pˢ))⁻¹
as well as Riemann's statement of his hypothesis and several others. Beineke and Hughes then define the moment of the modulus of the Riemann zeta function by
I_k(T)=1/T∫₀ ᪲|ζ(1/2+it)|²ᵏdt
and take us through the work of several mathematicians on properties of the second and fourth moments.

analytic continuation
analytic continuation

(number theory, analytic number theory, uncountable) The function ζ defined by the Dirichlet series ζ(s)=∑ₙ₌₁ ᪲1/(nˢ)=1/(1ˢ)+1/(2ˢ)+1/(3ˢ)+1/(4ˢ)+⋯, which is summable for points s in the complex half-plane with real part > 1; the analytic continuation of said function, being a holomorphic function defined on the complex numbers with pole at 1.

Riemann-zeta-functie

analytic continuation of a function defined as the sum of a Dirichlet series


priemgetal

head

priemgetal (plural priemgetallen, diminutive priemgetalletjie)

priemgetallen (plural)
priemgetalletjie (diminutive)


priemgetal

nl-noun

priemgetal n (plural priemgetallen, diminutive priemgetalletje n)

priemgetallen (plural)
priemgetalletje (diminutive neuter)

(number theory) prime number (natural number, only divisible by 1 and itself)


nombre premier

fr-noun

nombre premier m (plural nombres premiers)


Primzahl

de-noun

Primzahl f (genitive Primzahl, plural Primzahlen)

Primzahl (genitive)
Primzahlen (plural)
Primzahl (nominative singular)
Primzahlen (definite nominative plural)
Primzahl (genitive singular)
Primzahlen (definite genitive plural)
Primzahl (dative singular)
Primzahlen (dative definite plural)
Primzahl (accusative singular)
Primzahlen (accusative definite plural)


prímszám

prímszámok (plural)
prímszám (nominative singular)
prímszámok (nominative plural)
prímszámot (accusative singular)
prímszámokat (accusative plural)
prímszámnak (dative singular)
prímszámoknak (dative plural)
prímszámmal (instrumental singular)
prímszámokkal (instrumental plural)
prímszámért (causal-final singular)
prímszámokért (causal-final plural)
prímszámmá (singular translative)
prímszámokká (plural translative)
prímszámig (singular terminative)
prímszámokig (plural terminative)
prímszámként (essive-formal singular)
prímszámokként (essive-formal plural)
prímszámban (inessive singular)
prímszámokban (inessive plural)
prímszámon (singular superessive)
prímszámokon (plural superessive)
prímszámnál (adessive singular)
prímszámoknál (adessive plural)
prímszámba (illative singular)
prímszámokba (illative plural)
prímszámra (singular sublative)
prímszámokra (plural sublative)
prímszámhoz (allative singular)
prímszámokhoz (allative plural)
prímszámból (elative singular)
prímszámokból (elative plural)
prímszámról (delative singular)
prímszámokról (delative plural)
prímszámtól (ablative singular)
prímszámoktól (ablative plural)
prímszámé (singular)
prímszámoké (plural)
prímszáméi (singular)
prímszámokéi (plural)
prímszámom (first-person possessed-single possessive singular)
prímszámaim (first-person possessed-many possessive singular)
prímszámod (possessed-single possessive second-person singular)
prímszámaid (possessed-many possessive second-person singular)
prímszáma (possessed-single possessive singular third-person)
prímszámai (possessed-many possessive singular third-person)
prímszámunk (first-person plural possessed-single possessive)
prímszámaink (first-person plural possessed-many possessive)
prímszámotok (plural possessed-single possessive second-person)
prímszámaitok (plural possessed-many possessive second-person)
prímszámuk (plural possessed-single possessive third-person)
prímszámaik (plural possessed-many possessive third-person)


primtall

head

primtall n (definite singular primtallet, indefinite plural primtall, definite plural primtalla or primtallene)

primtallet (definite singular)
primtall (indefinite plural)
primtalla (definite plural)
primtallene (definite plural)

(number theory) a prime number (natural number)


liczba pierwsza

liczba pierwsza (nominative singular)
liczby pierwsze (nominative plural)
liczby pierwszej (genitive singular)
liczb pierwszych (genitive plural)
liczbie pierwszej (dative singular)
liczbom pierwszym (dative plural)
liczbę pierwszą (accusative singular)
liczby pierwsze (accusative plural)
liczbą pierwszą (instrumental singular)
liczbami pierwszymi (instrumental plural)
liczbie pierwszej (locative singular)
liczbach pierwszych (locative plural)
liczbo pierwsza (singular vocative)
liczby pierwsze (plural vocative)


número primo

(number theory) prime number (natural number)


простое число

ru-noun+

просто́е число́ • (prostóje čisló) n inan (genitive просто́го числа́, nominative plural просты́е чи́сла, genitive plural просты́х чи́сел)

просто́е число́ (canonical)
prostóje čisló (romanization)
просто́го числа́ (genitive)
просты́е чи́сла (nominative plural)
просты́х чи́сел (genitive plural)
просто́е число́ (nominative singular)
просты́е чи́сла (nominative plural)
просто́го числа́ (genitive singular)
просты́х чи́сел (genitive plural)
просто́му числу́ (dative singular)
просты́м чи́слам (dative plural)
просто́е число́ (accusative singular)
просты́е чи́сла (accusative plural)
просты́м число́м (instrumental singular)
просты́ми чи́слами (instrumental plural)
просто́м числе́ (prepositional singular)
просты́х чи́слах (plural prepositional)

ru:простое число


primtal

head

primtal n (singular definite primtallet, plural indefinite primtal)

primtallet (definite singular)
primtal (indefinite plural)
primtal (indefinite nominative singular)
primtallet (definite nominative singular)
primtal (indefinite nominative plural)
primtallene (definite nominative plural)
primtals (genitive indefinite singular)
primtallets (definite genitive singular)
primtals (genitive indefinite plural)
primtallenes (definite genitive plural)


primtal

primtal (indefinite nominative singular)
primtals (genitive indefinite singular)
primtalet (definite nominative singular)
primtalets (definite genitive singular)
primtal (indefinite nominative plural)
primtals (genitive indefinite plural)
primtalen (definite nominative plural)
primtalens (definite genitive plural)


asal sayı

asal sayı (definite accusative asal sayıyı, plural asal sayılar)

asal sayıyı (accusative definite)
asal sayılar (plural)


algebraic number theory

Blend of algebraic number + number theory. Possibly (alternatively or also) a calque of German algebraische Zahlentheorie.

blend

As is well-known in algebraic number theory, there is a far-reaching analogy between number fields and algebraic function fields in one variable with finite constant field.

1998, Norbert Klingen, Arithmetical Similarities: Prime Decomposition and Finite Group Theory, Oxford University Press (Clarendon Press), page 241:

The field of mathematics that these generalize to is known as algebraic number theory. As a matter of historical fact, our development here parallels the development of algebraic number theory.

2008, Steven H. Weintraub, Factorization: Unique and Otherwise, CRC Press, page 143:

Elementary arithmetic studies divisibility and factorization of ordinary integers, Algebraic number theory considers the same questions for algebraic numbers, solutions to polynomial equations with integer coefficients. In this setting, (un)fortunately, the uniqueness of prime factorization no longer holds. Remedying and measuring its failure is the starting point of algebraic number theory.

2013, Mak Trifković, Algebraic Theory of Quadratic Numbers, Springer, page v:

algebraic number field
algebraic number field

(mathematics, number theory) The branch of number theory in which number-theoretic questions are expressed in terms of properties of algebraic number fields or related objects, and studied using techniques from algebra.

հանրահաշվական թվերի տեսություն

branch of number theory concerning algebraic number fields


πρώτος αριθμός

head

πρώτος αριθμός

πρώτοι αριθμοί

πρώτος αριθμός • (prótos arithmós) m (plural πρώτοι αριθμοί)

el-noun

πρώτος αριθμός

πρώτοι αριθμοί

prótos arithmós (romanization)
πρώτοι αριθμοί (plural)


praštevilo

prȃštevílo (canonical neuter)
please help by adding the tonal marks.
práštevílo (nominative singular)
práštevíli (dual nominative)
práštevíla (nominative plural)
práštevíla (genitive singular)
práštevíl (dual genitive)
práštevíl (genitive plural)
práštevílu (dative singular)
práštevíloma (dative dual)
práštevílom (dative plural)
práštevílo (accusative singular)
práštevíli (accusative dual)
práštevíla (accusative plural)
práštevílu (locative singular)
práštevílih (dual locative)
práštevílih (locative plural)
práštevílom (instrumental singular)
práštevíloma (dual instrumental)
práštevíli (instrumental plural)

/práːʃtɛʋìːlɔ/


coprime

LL-Q1860 (eng)-Flame, not lame-coprime.wav

24 and 35 are coprime.

(number theory, of two or more positive integers) Having no positive integer factors in common, aside from 1.

взаимно прости

(of two or more positive integers) having no factors in common

keskenään jaoton ... kanssa

(of a positive integer) having no factor in common with...

24 is coprime to 35.

(number theory, of a positive integer) Having no positive integer factors, aside from 1, in common with one or more specified other positive integers.


простой

ru-adj

про́стенький

просто́й • (prostój) (comparative (по)про́ще, superlative просте́йший, diminutive про́стенький)

просто́й (canonical)
prostój (romanization)
(по)про́ще (comparative)
просте́йший (superlative)
про́стенький (diminutive)
просто́й (masculine nominative)
просто́е (neuter nominative)
проста́я (feminine nominative)
просты́е (nominative plural)
просто́го (genitive masculine neuter)
просто́й (feminine genitive)
просты́х (genitive plural)
просто́му (dative masculine neuter)
просто́й (dative feminine)
просты́м (dative plural)
просто́го (accusative animate masculine)
просто́е (accusative neuter)
просту́ю (accusative feminine)
просты́х (accusative animate plural)
просто́й (accusative inanimate masculine)
просты́е (accusative inanimate plural)
просты́м (instrumental masculine neuter)
просто́й (feminine instrumental)
просто́ю (feminine instrumental)
просты́ми (instrumental plural)
просто́м (masculine neuter prepositional)
просто́й (feminine prepositional)
просты́х (plural prepositional)
прост (masculine short-form)
про́сто (neuter short-form)
проста́ (feminine short-form)
просты́ (plural short-form)
про́сты (plural short-form)

→ Ingrian: prostoi

→ Yakut: боростуой (borostuoy)

Inherited from Old East Slavic простъ (prostŭ), from Proto-Slavic *prostъ.

inh+

Ru-простой.ogg

взаи́мно просты́е ― vzaímno prostýje ― coprime

просто́е число́


p-adic number

The expansion (21)2121ₚ is equal to the rational p-adic number #92;textstyle#123;2p#43;1#92;overp²-1#125;.

In the set of 3-adic numbers, the closed ball of radius 1/3 "centered" at 1, call it B, is the set x|∃n∈ℤ.,x=3n+1. This closed ball partitions into exactly three smaller closed balls of radius 1/9: x|∃n∈ℤ.,x=1+9n, x|∃n∈ℤ.,x=4+9n, and x|∃n∈ℤ.,x=7+9n. Then each of those balls partitions into exactly 3 smaller closed balls of radius 1/27, and the sub-partitioning can be continued indefinitely, in a fractal manner.
Likewise, going upwards in the hierarchy, B is part of the closed ball of radius 1 centered at 1, namely, the set of integers. Two other closed balls of radius 1 are "centered" at 1/3 and 2/3, and all three closed balls of radius 1 form a closed ball of radius 3, x|∃n∈ℤ.,x=1+n/3, which is one out of three closed balls forming a closed ball of radius 9, and so on.

3. In his recent book Professor Hensel has developed a theory of logarithms of the rational p'''-adic numbers, and from this he has shown how all such numbers can be written in the form p#92;alpha#92;omega#92;betae#92;gamma.

1914, Bulletin of the American Mathematical Society, page 452:

p'''-Adic numbers were introduced in mathematics by K. Hensel, and this invention led to substantial developments in number theory, where p'''-adic numbers are now as natural as ordinary real numbers.[…]Bleher noticed in [19] that the set of purely fractional p'''-adic numbers is an example of hierarchical lattice.

1991, M. D. Missarov, “Renormalization Group and Renormalization Theory in p-Adic and Adelic Scalar Models”, in Ya. G. Sinaĭ, editor, Dynamical Systems and Statistical Mechanics: From the Seminar on Statistical Physics held at Moscow State University, American Mathematical Society, page 143:

#92;Q#95;p is called the p-adic number field, and its elements are called p-adic numbers. In this section we introduce the p'''-adic number fields, which are very important objects in number theory.
The p'''-adic numbers were originally introduced by Hensel around 1900.

2000, Kazuya Kato, Nobushige Kurokawa, Takeshi Saitō, Takeshi Saito, translated by Masato Kuwata, Number Theory: Fermat's dream, American Mathematical Society, page 58:

(number theory) An element of a completion of the field of rational numbers with respect to a p-adic ultrametric.

p進數 /p进数

element of a completion of the rational numbers with respect to a p-adic ultrametric


záporné číslo

záporné číslo (nominative singular)
záporná čísla (nominative plural)
záporného čísla (genitive singular)
záporných čísel (genitive plural)
zápornému číslu (dative singular)
záporným číslům (dative plural)
záporné číslo (accusative singular)
záporná čísla (accusative plural)
záporné číslo (singular vocative)
záporná čísla (plural vocative)
záporném čísle (locative singular)
záporném číslu (locative singular)
záporných číslech (locative plural)
záporným číslem (instrumental singular)
zápornými čísly (instrumental plural)


magic cube

(number theory) By analogy with magic square, an n-by-n-by-n arrangement of n³ numbers such that the numbers in each row, in all columns (horizontal and vertical) and in all main diagonals (of which there are 6*n+4) each have the same sum.


composite number

en-noun

composite number (plural composite numbers)

It is evident, moreover, that if a composite number terminates in any one of the digits 1, 3, 7, 9, no one of its factors can be the number 5, or any number ending in 5.

1853, Edward Hinkley, Tables of the Prime Numbers and Prime Factors of the Composite Numbers from 1 to 100,000, page 13:

The above remark applies also to many composite numbers which are not Carmichael numbers, but which are still not shown to be composite by Fermat's theorem for a specific base a.[…]Therefore, using Euler's criterion or a strong primality test with enough bases will finally reveal any composite number or, alternatively, prove the primality of any prime N.

1994, Hans Riesel, Prime Numbers and Computer Methods for Factorization, 2nd edition, Springer (Birkhäuser), →ISBN, page 91:

2006, Sze Kui Ng, Quantum Invariant of 3-Manifolds, Samuel F. Neilson (editor), New Research on Three-manifolds and Mathematics, Nova Science Publishers, page 47,
It is interesting that in each interval composite numbers are one-to-one assigned to the comparable composite knots while prime numbers are one-to-one assigned to prime knots.

(number theory) A (nonzero) natural number that is expressible as the product of two (or more) natural numbers other than itself and 1.

عَدَد غَيْر أَوَّلِيّ

number that is the product of two natural numbers other than itself and 1


amicable number

amicable number (plural amicable numbers)

The factors of 220, excluding 220 itself, are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110 and sum to 284; the factors of 284, excluding 284 itself, are 1, 2, 4, 71, and 142, and sum to 220; hence 220 and 284 are amicable numbers.

(number theory) One of a pair of numbers having the property that the sum of the divisors of each, excluding itself, is equal to the other number.


harmonic number

harmonic number (plural harmonic numbers)

The fourth harmonic number has a value of H_3=∑ₖ₌₁³1/k=1/1+1/2+1/3=1 5/6.

(number theory) Any of a series of numbers formed from the sum of the reciprocals of consecutive natural numbers


figurate number

figurate number (plural figurate numbers)

(mathematics, number theory) A number equal to the number of dots in a geometric figure formed of dots. The series begins with unity and is formed so that subtracting each from the following, and treating the series so formed in the same way, by a continuation of the process, yields equal differences.


semiprime

Again, to be perfectly clear, we are looking for c values that produce a low density of semiprimes when employing Euler's basic polynomial but changing the c values, in the range of x=1 to 10000. Some very early standouts are: c=4 which produces 799 semiprimes; c=6 which produces 532 semiprimes; c=12 which produces only 431 semiprimes; c=18 which produces 364 semiprimes, and c=30 which produces only 320 semiprimes.

2010, Jason Earls, The Lowbrow Experimental Mathematician, Lulu.com, page 145:

2015, Jie Wang, Zachary A. Kissel, Introduction to Network Security: Theory and Practice, Wiley [under licence from Higher Education Press], page 113,
Firstly, we should change semiprimes from time to time, where a particular semiprime should only be used in a time interval shorter than the time required to factor an RSA challenge number of a similar length. Secondly, we should use semiprimes that consist of more than 200 decimal digits.

In this paper I will define four sequences of numbers obtained through concatenation, definitions which also use the notion of “sum of the digits of a number”, sequences that have the property to produce many primes, semiprimes and products of very few prime factors.

2015, Marius Coman, Two Hundred and Thirteen Conjectures on Primes: Collected Papers, Education Publishing, page 46:

(number theory) A natural number that is the product of two (not necessarily distinct) prime numbers.

puolialkuluku

number that is the product of two primes

biprime

product of two primes, not necessarily distinct


rationnel

fr-adj

rationnel (feminine rationnelle, masculine plural rationnels, feminine plural rationnelles)

rationnelle (feminine)
rationnels (masculine plural)
rationnelles (feminine plural)

Learned borrowing from Latin ratiōnālis.

(mathematics, number theory, arithmetic) rational (of a number, capable of being expressed as the ratio of two integers)


runsas luku

runsas luku (nominative singular)
runsaat luvut (nominative plural)
runsas luku (accusative nominative singular)
runsaat luvut (accusative nominative plural)
runsaan luvun (accusative genitive singular)
runsaat luvut (accusative genitive plural)
runsaan luvun (genitive singular)
runsaiden lukujen (genitive plural)
runsaitten lukujen (genitive plural rare)
runsasten lukujen (genitive plural rare)
runsasta lukua (partitive singular)
runsaita lukuja (partitive plural)
runsaassa luvussa (inessive singular)
runsaissa luvuissa (inessive plural)
runsaasta luvusta (elative singular)
runsaista luvuista (elative plural)
runsaaseen lukuun (illative singular)
runsaisiin lukuihin (illative plural)
runsaihin lukuihin (illative plural rare)
runsaalla luvulla (adessive singular)
runsailla luvuilla (adessive plural)
runsaalta luvulta (ablative singular)
runsailta luvuilta (ablative plural)
runsaalle luvulle (allative singular)
runsaille luvuille (allative plural)
runsaana lukuna (essive singular)
runsaina lukuina (essive plural)
runsaaksi luvuksi (singular translative)
runsaiksi luvuiksi (plural translative)
runsaatta luvutta (abessive singular)
runsaitta luvuitta (abessive plural)
runsain luvuin (instructive plural)
See the possessive forms below. (comitative)
runsas lukuni (nominative singular)
runsaat lukuni (nominative plural)
runsas lukuni (accusative nominative singular)
runsaat lukuni (accusative nominative plural)
runsaan lukuni (accusative genitive singular)
runsaat lukuni (accusative genitive plural)
runsaan lukuni (genitive singular)
runsaiden lukujeni (genitive plural)
runsaitten lukujeni (genitive plural rare)
runsasten lukujeni (genitive plural rare)
runsasta lukuani (partitive singular)
runsaita lukujani (partitive plural)
runsaassa luvussani (inessive singular)
runsaissa luvuissani (inessive plural)
runsaasta luvustani (elative singular)
runsaista luvuistani (elative plural)
runsaaseen lukuuni (illative singular)
runsaisiin lukuihini (illative plural)
runsaihin lukuihini (illative plural rare)
runsaalla luvullani (adessive singular)
runsailla luvuillani (adessive plural)
runsaalta luvultani (ablative singular)
runsailta luvuiltani (ablative plural)
runsaalle luvulleni (allative singular)
runsaille luvuilleni (allative plural)
runsaana lukunani (essive singular)
runsaina lukuinani (essive plural)
runsaaksi luvukseni (singular translative)
runsaiksi luvuikseni (plural translative)
runsaatta luvuttani (abessive singular)
runsaitta luvuittani (abessive plural)
runsaine lukuineni (comitative plural)
runsas lukusi (nominative singular)
runsaat lukusi (nominative plural)
runsas lukusi (accusative nominative singular)
runsaat lukusi (accusative nominative plural)
runsaan lukusi (accusative genitive singular)
runsaat lukusi (accusative genitive plural)
runsaan lukusi (genitive singular)
runsaiden lukujesi (genitive plural)
runsaitten lukujesi (genitive plural rare)
runsasten lukujesi (genitive plural rare)
runsasta lukuasi (partitive singular)
runsaita lukujasi (partitive plural)
runsaassa luvussasi (inessive singular)
runsaissa luvuissasi (inessive plural)
runsaasta luvustasi (elative singular)
runsaista luvuistasi (elative plural)
runsaaseen lukuusi (illative singular)
runsaisiin lukuihisi (illative plural)
runsaihin lukuihisi (illative plural rare)
runsaalla luvullasi (adessive singular)
runsailla luvuillasi (adessive plural)
runsaalta luvultasi (ablative singular)
runsailta luvuiltasi (ablative plural)
runsaalle luvullesi (allative singular)
runsaille luvuillesi (allative plural)
runsaana lukunasi (essive singular)
runsaina lukuinasi (essive plural)
runsaaksi luvuksesi (singular translative)
runsaiksi luvuiksesi (plural translative)
runsaatta luvuttasi (abessive singular)
runsaitta luvuittasi (abessive plural)
runsaine lukuinesi (comitative plural)
runsas lukumme (nominative singular)
runsaat lukumme (nominative plural)
runsas lukumme (accusative nominative singular)
runsaat lukumme (accusative nominative plural)
runsaan lukumme (accusative genitive singular)
runsaat lukumme (accusative genitive plural)
runsaan lukumme (genitive singular)
runsaiden lukujemme (genitive plural)
runsaitten lukujemme (genitive plural rare)
runsasten lukujemme (genitive plural rare)
runsasta lukuamme (partitive singular)
runsaita lukujamme (partitive plural)
runsaassa luvussamme (inessive singular)
runsaissa luvuissamme (inessive plural)
runsaasta luvustamme (elative singular)
runsaista luvuistamme (elative plural)
runsaaseen lukuumme (illative singular)
runsaisiin lukuihimme (illative plural)
runsaihin lukuihimme (illative plural rare)
runsaalla luvullamme (adessive singular)
runsailla luvuillamme (adessive plural)
runsaalta luvultamme (ablative singular)
runsailta luvuiltamme (ablative plural)
runsaalle luvullemme (allative singular)
runsaille luvuillemme (allative plural)
runsaana lukunamme (essive singular)
runsaina lukuinamme (essive plural)
runsaaksi luvuksemme (singular translative)
runsaiksi luvuiksemme (plural translative)
runsaatta luvuttamme (abessive singular)
runsaitta luvuittamme (abessive plural)
runsaine lukuinemme (comitative plural)
runsas lukunne (nominative singular)
runsaat lukunne (nominative plural)
runsas lukunne (accusative nominative singular)
runsaat lukunne (accusative nominative plural)
runsaan lukunne (accusative genitive singular)
runsaat lukunne (accusative genitive plural)
runsaan lukunne (genitive singular)
runsaiden lukujenne (genitive plural)
runsaitten lukujenne (genitive plural rare)
runsasten lukujenne (genitive plural rare)
runsasta lukuanne (partitive singular)
runsaita lukujanne (partitive plural)
runsaassa luvussanne (inessive singular)
runsaissa luvuissanne (inessive plural)
runsaasta luvustanne (elative singular)
runsaista luvuistanne (elative plural)
runsaaseen lukuunne (illative singular)
runsaisiin lukuihinne (illative plural)
runsaihin lukuihinne (illative plural rare)
runsaalla luvullanne (adessive singular)
runsailla luvuillanne (adessive plural)
runsaalta luvultanne (ablative singular)
runsailta luvuiltanne (ablative plural)
runsaalle luvullenne (allative singular)
runsaille luvuillenne (allative plural)
runsaana lukunanne (essive singular)
runsaina lukuinanne (essive plural)
runsaaksi luvuksenne (singular translative)
runsaiksi luvuiksenne (plural translative)
runsaatta luvuttanne (abessive singular)
runsaitta luvuittanne (abessive plural)
runsaine lukuinenne (comitative plural)
runsas lukunsa (nominative singular)
runsaat lukunsa (nominative plural)
runsas lukunsa (accusative nominative singular)
runsaat lukunsa (accusative nominative plural)
runsaan lukunsa (accusative genitive singular)
runsaat lukunsa (accusative genitive plural)
runsaan lukunsa (genitive singular)
runsaiden lukujensa (genitive plural)
runsaitten lukujensa (genitive plural rare)
runsasten lukujensa (genitive plural rare)
runsasta lukuaan (partitive singular)
runsasta lukuansa (partitive singular)
runsaita lukujaan (partitive plural)
runsaita lukujansa (partitive plural)
runsaassa luvussaan (inessive singular)
runsaassa luvussansa (inessive singular)
runsaissa luvuissaan (inessive plural)
runsaissa luvuissansa (inessive plural)
runsaasta luvustaan (elative singular)
runsaasta luvustansa (elative singular)
runsaista luvuistaan (elative plural)
runsaista luvuistansa (elative plural)
runsaaseen lukuunsa (illative singular)
runsaisiin lukuihinsa (illative plural)
runsaihin lukuihinsa (illative plural rare)
runsaalla luvullaan (adessive singular)
runsaalla luvullansa (adessive singular)
runsailla luvuillaan (adessive plural)
runsailla luvuillansa (adessive plural)
runsaalta luvultaan (ablative singular)
runsaalta luvultansa (ablative singular)
runsailta luvuiltaan (ablative plural)
runsailta luvuiltansa (ablative plural)
runsaalle luvulleen (allative singular)
runsaalle luvullensa (allative singular)
runsaille luvuilleen (allative plural)
runsaille luvuillensa (allative plural)
runsaana lukunaan (essive singular)
runsaana lukunansa (essive singular)
runsaina lukuinaan (essive plural)
runsaina lukuinansa (essive plural)
runsaaksi luvukseen (singular translative)
runsaaksi luvuksensa (singular translative)
runsaiksi luvuikseen (plural translative)
runsaiksi luvuiksensa (plural translative)
runsaatta luvuttaan (abessive singular)
runsaatta luvuttansa (abessive singular)
runsaitta luvuittaan (abessive plural)
runsaitta luvuittansa (abessive plural)
runsaine lukuineen (comitative plural)
runsaine lukuinensa (comitative plural)


Proth number

After French mathematician François Proth (1852-1879).

2006, B. Grégoire, L. Théry, B. Werner, A Computational Approach to Pocklington Certificates, Masami Hagiya, Philip Wadler (editors), Functional and Logic Programming: 8th International Symposium, Proceedings, Springer, LNCS 3945, page 109,
To generate Pocklington certificates for Proth number we add a new entry to the oracle: pocklington -proth k p.

Suppose that a Proth number n#61;k2ʳ#43;1 satisfies the condition that a#123;(n-1)#47;2#125;#92;equiv-1#92;pmodn for some integer a. Prove that n is prime.

2016, Abhijit Das, Computational Number Theory, Taylor & Francis (CRC Press / Chapman & Hall), page 295:

If a Proth number is prime, we call it a Proth prime.

2014, Adam Spencer, Adam Spencer's Big Book of Numbers, Brio Books, page 388:

(number theory) Any number of the form k·2ⁿ + 1, where k is odd, n is a positive integer, and 2ⁿ > k.


adjunction

en-noun

adjunction (countable and uncountable, plural adjunctions)

LL-Q1860 (eng)-Flame, not lame-adjunction.wav

From Latin adjunctio, from adjungere: compare French adjonction, and see adjunct.

The ring obtained after the adjunction of the elements a,b and y to the ring R may be denoted R#91;a,b,y#93;.

The field adjunction #92;mathbb#123;Q#125;(#92;pi) can be obtained from #92;mathbb#123;Q#125; by adjoining #92;pi to #92;mathbb#123;Q#125;.

(mathematics, chiefly algebra and number theory) The process of adjoining elements to an algebraic structure (usually a ring or field); the result of such a process.


պարզ թիվ

head

պարզ թիվ • (parz tʻiv)

hy-noun

պարզ թիվ • (parz tʻiv)


täydellinen luku

täydellinen luku (nominative singular)
täydelliset luvut (nominative plural)
täydellinen luku (accusative nominative singular)
täydelliset luvut (accusative nominative plural)
täydellisen luvun (accusative genitive singular)
täydelliset luvut (accusative genitive plural)
täydellisen luvun (genitive singular)
täydellisten lukujen (genitive plural)
täydellisien lukujen (genitive plural)
täydellistä lukua (partitive singular)
täydellisiä lukuja (partitive plural)
täydellisessä luvussa (inessive singular)
täydellisissä luvuissa (inessive plural)
täydellisestä luvusta (elative singular)
täydellisistä luvuista (elative plural)
täydelliseen lukuun (illative singular)
täydellisiin lukuihin (illative plural)
täydellisellä luvulla (adessive singular)
täydellisillä luvuilla (adessive plural)
täydelliseltä luvulta (ablative singular)
täydellisiltä luvuilta (ablative plural)
täydelliselle luvulle (allative singular)
täydellisille luvuille (allative plural)
täydellisenä lukuna (essive singular)
täydellisinä lukuina (essive plural)
täydelliseksi luvuksi (singular translative)
täydellisiksi luvuiksi (plural translative)
täydellisettä luvutta (abessive singular)
täydellisittä luvuitta (abessive plural)
täydellisin luvuin (instructive plural)
See the possessive forms below. (comitative)
täydellinen lukuni (nominative singular)
täydelliset lukuni (nominative plural)
täydellinen lukuni (accusative nominative singular)
täydelliset lukuni (accusative nominative plural)
täydellisen lukuni (accusative genitive singular)
täydelliset lukuni (accusative genitive plural)
täydellisen lukuni (genitive singular)
täydellisten lukujeni (genitive plural)
täydellisien lukujeni (genitive plural)
täydellistä lukuani (partitive singular)
täydellisiä lukujani (partitive plural)
täydellisessä luvussani (inessive singular)
täydellisissä luvuissani (inessive plural)
täydellisestä luvustani (elative singular)
täydellisistä luvuistani (elative plural)
täydelliseen lukuuni (illative singular)
täydellisiin lukuihini (illative plural)
täydellisellä luvullani (adessive singular)
täydellisillä luvuillani (adessive plural)
täydelliseltä luvultani (ablative singular)
täydellisiltä luvuiltani (ablative plural)
täydelliselle luvulleni (allative singular)
täydellisille luvuilleni (allative plural)
täydellisenä lukunani (essive singular)
täydellisinä lukuinani (essive plural)
täydelliseksi luvukseni (singular translative)
täydellisiksi luvuikseni (plural translative)
täydellisettä luvuttani (abessive singular)
täydellisittä luvuittani (abessive plural)
täydellisine lukuineni (comitative plural)
täydellinen lukusi (nominative singular)
täydelliset lukusi (nominative plural)
täydellinen lukusi (accusative nominative singular)
täydelliset lukusi (accusative nominative plural)
täydellisen lukusi (accusative genitive singular)
täydelliset lukusi (accusative genitive plural)
täydellisen lukusi (genitive singular)
täydellisten lukujesi (genitive plural)
täydellisien lukujesi (genitive plural)
täydellistä lukuasi (partitive singular)
täydellisiä lukujasi (partitive plural)
täydellisessä luvussasi (inessive singular)
täydellisissä luvuissasi (inessive plural)
täydellisestä luvustasi (elative singular)
täydellisistä luvuistasi (elative plural)
täydelliseen lukuusi (illative singular)
täydellisiin lukuihisi (illative plural)
täydellisellä luvullasi (adessive singular)
täydellisillä luvuillasi (adessive plural)
täydelliseltä luvultasi (ablative singular)
täydellisiltä luvuiltasi (ablative plural)
täydelliselle luvullesi (allative singular)
täydellisille luvuillesi (allative plural)
täydellisenä lukunasi (essive singular)
täydellisinä lukuinasi (essive plural)
täydelliseksi luvuksesi (singular translative)
täydellisiksi luvuiksesi (plural translative)
täydellisettä luvuttasi (abessive singular)
täydellisittä luvuittasi (abessive plural)
täydellisine lukuinesi (comitative plural)
täydellinen lukumme (nominative singular)
täydelliset lukumme (nominative plural)
täydellinen lukumme (accusative nominative singular)
täydelliset lukumme (accusative nominative plural)
täydellisen lukumme (accusative genitive singular)
täydelliset lukumme (accusative genitive plural)
täydellisen lukumme (genitive singular)
täydellisten lukujemme (genitive plural)
täydellisien lukujemme (genitive plural)
täydellistä lukuamme (partitive singular)
täydellisiä lukujamme (partitive plural)
täydellisessä luvussamme (inessive singular)
täydellisissä luvuissamme (inessive plural)
täydellisestä luvustamme (elative singular)
täydellisistä luvuistamme (elative plural)
täydelliseen lukuumme (illative singular)
täydellisiin lukuihimme (illative plural)
täydellisellä luvullamme (adessive singular)
täydellisillä luvuillamme (adessive plural)
täydelliseltä luvultamme (ablative singular)
täydellisiltä luvuiltamme (ablative plural)
täydelliselle luvullemme (allative singular)
täydellisille luvuillemme (allative plural)
täydellisenä lukunamme (essive singular)
täydellisinä lukuinamme (essive plural)
täydelliseksi luvuksemme (singular translative)
täydellisiksi luvuiksemme (plural translative)
täydellisettä luvuttamme (abessive singular)
täydellisittä luvuittamme (abessive plural)
täydellisine lukuinemme (comitative plural)
täydellinen lukunne (nominative singular)
täydelliset lukunne (nominative plural)
täydellinen lukunne (accusative nominative singular)
täydelliset lukunne (accusative nominative plural)
täydellisen lukunne (accusative genitive singular)
täydelliset lukunne (accusative genitive plural)
täydellisen lukunne (genitive singular)
täydellisten lukujenne (genitive plural)
täydellisien lukujenne (genitive plural)
täydellistä lukuanne (partitive singular)
täydellisiä lukujanne (partitive plural)
täydellisessä luvussanne (inessive singular)
täydellisissä luvuissanne (inessive plural)
täydellisestä luvustanne (elative singular)
täydellisistä luvuistanne (elative plural)
täydelliseen lukuunne (illative singular)
täydellisiin lukuihinne (illative plural)
täydellisellä luvullanne (adessive singular)
täydellisillä luvuillanne (adessive plural)
täydelliseltä luvultanne (ablative singular)
täydellisiltä luvuiltanne (ablative plural)
täydelliselle luvullenne (allative singular)
täydellisille luvuillenne (allative plural)
täydellisenä lukunanne (essive singular)
täydellisinä lukuinanne (essive plural)
täydelliseksi luvuksenne (singular translative)
täydellisiksi luvuiksenne (plural translative)
täydellisettä luvuttanne (abessive singular)
täydellisittä luvuittanne (abessive plural)
täydellisine lukuinenne (comitative plural)
täydellinen lukunsa (nominative singular)
täydelliset lukunsa (nominative plural)
täydellinen lukunsa (accusative nominative singular)
täydelliset lukunsa (accusative nominative plural)
täydellisen lukunsa (accusative genitive singular)
täydelliset lukunsa (accusative genitive plural)
täydellisen lukunsa (genitive singular)
täydellisten lukujensa (genitive plural)
täydellisien lukujensa (genitive plural)
täydellistä lukuaan (partitive singular)
täydellistä lukuansa (partitive singular)
täydellisiä lukujaan (partitive plural)
täydellisiä lukujansa (partitive plural)
täydellisessä luvussaan (inessive singular)
täydellisessä luvussansa (inessive singular)
täydellisissä luvuissaan (inessive plural)
täydellisissä luvuissansa (inessive plural)
täydellisestä luvustaan (elative singular)
täydellisestä luvustansa (elative singular)
täydellisistä luvuistaan (elative plural)
täydellisistä luvuistansa (elative plural)
täydelliseen lukuunsa (illative singular)
täydellisiin lukuihinsa (illative plural)
täydellisellä luvullaan (adessive singular)
täydellisellä luvullansa (adessive singular)
täydellisillä luvuillaan (adessive plural)
täydellisillä luvuillansa (adessive plural)
täydelliseltä luvultaan (ablative singular)
täydelliseltä luvultansa (ablative singular)
täydellisiltä luvuiltaan (ablative plural)
täydellisiltä luvuiltansa (ablative plural)
täydelliselle luvulleen (allative singular)
täydelliselle luvullensa (allative singular)
täydellisille luvuilleen (allative plural)
täydellisille luvuillensa (allative plural)
täydellisenä lukunaan (essive singular)
täydellisenä lukunansa (essive singular)
täydellisinä lukuinaan (essive plural)
täydellisinä lukuinansa (essive plural)
täydelliseksi luvukseen (singular translative)
täydelliseksi luvuksensa (singular translative)
täydellisiksi luvuikseen (plural translative)
täydellisiksi luvuiksensa (plural translative)
täydellisettä luvuttaan (abessive singular)
täydellisettä luvuttansa (abessive singular)
täydellisittä luvuittaan (abessive plural)
täydellisittä luvuittansa (abessive plural)
täydellisine lukuineen (comitative plural)
täydellisine lukuinensa (comitative plural)


მარტივი რიცხვი

ka-noun

მარტივი რიცხვი • (marṭivi ricxvi) (plural მარტივი რიცხვები)

marṭivi ricxvi (romanization)
მარტივი რიცხვები (plural)


zenbaki lehen

zenbaki lehen (absolutive indefinite)
zenbaki lehena (absolutive indefinite singular)
zenbaki lehenak (absolutive indefinite plural)
zenbaki lehenek (ergative indefinite)
zenbaki lehenak (ergative indefinite singular)
zenbaki lehenek (ergative indefinite plural)
zenbaki leheni (dative indefinite)
zenbaki lehenari (dative indefinite singular)
zenbaki lehenei (dative indefinite plural)
zenbaki lehenen (genitive indefinite)
zenbaki lehenaren (genitive indefinite singular)
zenbaki lehenen (genitive indefinite plural)
zenbaki lehenekin (comitative indefinite)
zenbaki lehenarekin (comitative indefinite singular)
zenbaki lehenekin (comitative indefinite plural)
zenbaki lehenengatik (causative indefinite)
zenbaki lehenarengatik (causative indefinite singular)
zenbaki lehenengatik (causative indefinite plural)
zenbaki lehenentzat (benefactive indefinite)
zenbaki lehenarentzat (benefactive indefinite singular)
zenbaki lehenentzat (benefactive indefinite plural)
zenbaki lehenez (indefinite instrumental)
zenbaki lehenaz (indefinite instrumental singular)
zenbaki lehenez (indefinite instrumental plural)
zenbaki lehenetan (indefinite inessive)
zenbaki lehenean (indefinite inessive singular)
zenbaki lehenetan (indefinite inessive plural)
zenbaki lehenetako (indefinite locative)
zenbaki leheneko (indefinite locative singular)
zenbaki lehenetako (indefinite locative plural)
zenbaki lehenetara (allative indefinite)
zenbaki lehenera (allative indefinite singular)
zenbaki lehenetara (allative indefinite plural)
zenbaki lehenetaraino (indefinite terminative)
zenbaki leheneraino (indefinite singular terminative)
zenbaki lehenetaraino (indefinite plural terminative)
zenbaki lehenetarantz (directive indefinite)
zenbaki lehenerantz (directive indefinite singular)
zenbaki lehenetarantz (directive indefinite plural)
zenbaki lehenetarako (destinative indefinite)
zenbaki lehenerako (destinative indefinite singular)
zenbaki lehenetarako (destinative indefinite plural)
zenbaki lehenetatik (ablative indefinite)
zenbaki lehenetik (ablative indefinite singular)
zenbaki lehenetatik (ablative indefinite plural)
zenbaki lehenik (indefinite partitive)
zenbaki lehentzat (indefinite prolative)

From zenbaki (“number”) + lehen (“prime, first”).


divisibility sequence

en-noun

divisibility sequence (plural divisibility sequences)

If #92;mathcalC#61;(C#95;n)#95;#123;n#92;ge 1#125; is any divisibility sequence, one says that a prime p is a primitive divisor of C#95;n if p#92;vertC#95;n but p#92;nmidC#95;1C#95;2C#95;#123;n-1#125;. Primitive divisors of certain divisibility sequences were studied by Zsigmondy [37] in the 19ᵗʰ century.

2012, P. Ingram, J. H. Silverman, “Primitive Divisors in Elliptic Divisibility Sequences”, in Dorian Goldfeld, Jay Jorgenson, Peter Jones, Dinakar Ramakrishnan, Kenneth Ribet, John Tate, editors, Number Theory, Analysis and Geometry, Springer,, page 244:

These divisibility sequences satisfy the same recurrence relations as the polynomials #92;phi#95;n and #92;psi²#95;n (see Appendix C).

2013, Graham Everest, Thomas Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Springer, page 138:

Moreover, in order to keep a divisibility sequence normalized, we can assume without loss of generality that a#95;0 = 0 and a#95;1#61;1.

2021, Masum Billal, Samin Riasat, Integer Sequences, Springer, page 59:

(number theory, algebra) Any sequence of integers {aₙ}, indexed by the natural numbers, such that if n is divisible by m then aₙ is divisible by aₘ.


premiers entre eux

fr-adj

premiers entre eux (plural only, feminine plural premières entre elles)

3 et 5 sont premiers entre eux.

(number theory) relatively prime, coprime (having no positive integer factors in common aside from 1)


nguyên

LL-Q9199 (vie)-Penn Zero MSSJ-nguyên.wav

nguyên thuỷ

(mythology, religion) primordial; primitive

số nguyên

phần nguyên


bűvös négyzet

(number theory) magic square (an n-by-n arrangement of n² numbers such that the numbers in each row, in each column and along both diagonals all have the same sum)


newform

(mathematics, number theory) A cusp form that is "new" at a given level N, where the levels are the nested subgroups Γ₀(N) of the modular group, with N ordered by divisibility.


samleifa

From sam- (“co-, fellow”) + leif (“deviation from regression, remainder, remainder term”).


uimhir phríomha

head

uimhir phríomha f (genitive singular uimhreach príomha, nominative plural uimhreacha príomha)

uimhreach príomha (genitive singular)
uimhreacha príomha (nominative plural)
nominative (indefinite)
uimhir phríomha (indefinite singular)
uimhreacha príomha (indefinite plural)
vocative (indefinite)
a uimhir phríomha (indefinite singular)
a uimhreacha príomha (indefinite plural)
genitive (indefinite)
uimhreach príomha (indefinite singular)
dative (indefinite)
nominative (definite)
an uimhir phríomha (definite singular)
na huimhreacha príomha (definite plural)
genitive (definite)
na huimhreach príomha (definite singular)
na n-uimhreacha príomha (definite plural)
dative (definite)
leis an uimhir phríomha (definite singular)
don uimhir phríomha (definite singular)
leis na huimhreacha príomha (definite plural)
uimhir phríomha
n-uimhir phríomha
huimhir phríomha


ard-earroo

ard-earroo m (genitive singular [please provide], plural [please provide])


primorial

Blend of prime + factorial. Coined by American engineer and mathematician Harvey Dubner.

For example, up to 10⁹, every positive integer which is a multiple of a primorial and is less than the following primorial satisfies the given inequality. There has also been little discussion on which primorials qualify in the different situations when RH fails.

2017, Kevin Broughan, Equivalents of the Riemann Hypothesis: Volume 1, Arithmetic Equivalents, Cambridge University Press, page 97:

(number theory) Any number belonging to the integer sequence whose nth element is the product of the first n primes.

The primorial of 6 is #92;textstyle 6#92;#35;#61;2#92;times 3#92;times 5#61;30.

2020, Rong Pan, Qinheping Hu, Rishabh Singh, Loris D'Antoni, Solving Problem Sketches with Large Integer Values, Peter Müller (editor), Programming Languages and Systems: 29th European Symposium, Proceedings, Springer, LNCS 12075, page 587,
The following number theory result relates the primorial to the Chebyshev function.
θ(n)= log (n#)= log 2⁽¹⁺ᵒ⁽ⁿ⁾⁾ⁿ=(1+o(n))n

(number theory) A unary operation, denoted by the postfix symbol # and defined on the nonnegative integers, which maps 0 to 1, 1 to 1, and each subsequent number to the product of all primes less than or equal to it; the value mapped to by said operation for a given input.


weird number

(number theory) A natural number that is abundant but not semiperfect; one for which the sum of the proper divisors (including 1 but not itself) is greater than the number, but no subset of those divisors sums to the number itself.


least common multiple

en-noun

least common multiple (plural least common multiples)

120 is the least common multiple of 60, 8 and 15.

The LEAST COMMON MULTIPLE is the smallest number, which can be measured by each of two, or more divisors.
[…]
Since 0 is the least multiple of every number, it must be the least common multiple of any, or of all numbers. It is, however, excluded by common consent in finding the least common multiples of numbers, because as such, 0 has no practical value.

1859, David Henry Cruttenden, The Objective Or Synthetic Arithmetic, J. M. Bradstreet & Son, page 261:

The least common multiple (LCM) is the smallest common multiple of two or more factors.[…]The least common multiple is used to determine the appropriate denominator for adding unlike fractions. In adding unlike fractions, it is important to find the lowest common denominator of the fractions. This is exactly the same thing as finding the least common multiple of the denominators.

2000, Robert W. Smith, Math Challenges, Grades 4-6, Teacher Created Resources, page 75:

It is sometimes necessary to determine the smallest common nonzero multiple of two or more whole numbers. We call this nonzero multiple the least common multiple. In our work with fractions, there will be problems where it will be necessary to find the least common multiple of some numbers, usually the denominators of fractions.

2007, Jerome Kaufmann, Karen Schwitters, Algebra for College Students, 8th edition, Thomson Learning, page 820:

(number theory) The smallest positive integer which is divisible by (equivalently, is an integer multiple of) each of a specified finite set of integers.


imaginary unit

So named because it takes on the role of unit for the imaginary part of a complex number.

In Section 4.3 we introduced the imaginary unit i with (i)²#61;-1, in the complex space domain. In section 4.4 we introduced the complex unit j with (j)²#61;-1, in connection with the time dependency of the shape (4.4.16) of the swimming profile. These different imaginary units have no interaction, we leave unaltered the product ij.

2013, J. A. Sparenberg, Hydrodynamic Propulsion and Its Optimization, Springer, page 183:

2014, Dennis G. Zill, Warren S. Wright, Advanced Engineering Mathematics, Ascend Learning (Jones & Bartlett Learning), 5th Edition, page 793,
We now simply say that i is the imaginary unit and define it by the property i²=-1. Using the imaginary unit, we build a general complex number out of two real numbers.

Consider now in addition to the given #92;H#92;subset#92;mathbb#123;O#125;, one of the totally imaginary units not contained in #92;H, and note that #123;SU#125;(2) acts transitively on the set of totally imaginary units of a quaternion algebra; as we are assuming #92;alpha#92;notin#92;H, we may in fact assume that #92;mathbb#123;O#125;#61;#92;H#92;oplus#92;H' with #92;alpha#92;in#92;H'.

2021, Bruce Hunt, Locally Mixed Symmetric Spaces, Springer, page 123:

(number theory, complex analysis, quaternion theory) An imaginary number (in the case of complex numbers, usually denoted i) that is defined as a solution to the equation x²=-1.


Landau-Ramanujan constant

After its two independent discoverers, Edmund Landau and Srinivasa Ramanujan.

(number theory) The constant of proportionality (approximately 0.7642) in the relationship between the number of positive integers less than x that are the sum of two square numbers, for large x, and the expression x/√.


прости број

sh-noun

прости број m (Latin spelling prosti broj)


p-adic absolute value

en-noun

p-adic absolute value (plural p-adic absolute values)

According to Ostrowski's theorem, only three kinds of norms are possible for the set of real numbers: the trivial absolute value, the real absolute value, and the p-adic absolute value.ᵂᴾ

If cgt;1, then x#92;rightarrowc#123;-v#95;p(x)#125; (with the convention c#123;-#92;infty#125;#61;0) is a nonarchimedean absolute value, denoted #92;vert..#92;vert#95;#123;p,c#125; and called the p-adic absolute value to base c. If cgt;1 and dgt;1 and if r#61;#92;log#95;cd, then #92;vertx#92;vert#95;#123;p,d#125;#61;#92;vertx#92;vertʳ#95;#123;p,c#125; for every x#92;inK. The p-adic topology on K is the topology defined by the p'''-adic absolute values.

1993, Seth Warner, Topological Rings, Elsevier (North-Holland), page 8:

They both gave essentially the same proof, based on the Subspace Theorem (more precisely, Schlickewei's generalisation to p'''-adic absolute values and number fields [30] of the Subspace Theorem proved by Schmidt in 1972 [41]).

1999, Jan-Hendrik Evertse, Hans Peter Schlickewei, “The Absolute Subspace Theorem and linear equations with unknowns from a multiplicative group”, in Kálmán Györy, Henryk Iwaniec, Jerzy Urbanowicz, editors, Number Theory in Progress, Walter de Gruyter, page 121:

It ^([#92;Z#95;p]) can also be defined as the subset ^([of #92;Q#95;p]) with #92;vertx#92;vert#95;plt;p because the p'''-adic absolute value takes no values between 1 and p, and therefore #92;Z#95;p is open.

2007, Anthony W. Knapp, Advanced Algebra, Springer (Birkhäuser), page 320:

(number theory, field theory) A norm for the rational numbers, with some prime number p as parameter, such that any rational number of the form pᵏ(a/b) — where a, b and k are integers and a, b and p are coprime — is mapped to the rational number p⁻ᵏ and 0 is mapped to 0. (Note: any nonzero rational number can be reduced to such a form.)


p-adic ordinal

Notice the resemblance between the p-adic ordinal and the base-p logarithm.

(number theory) A function of rational numbers, with prime number p as parameter, which is defined for some non-zero integer x as the largest integer r such that pʳ divides x; is defined for some non-zero rational number a/b as the p-adic ordinal of a minus the p-adic ordinal of b; and is defined for 0 as infinity.


p-adic norm

2002, M. Ram Murty, Introduction to p-adic Analytic Number Theory, American Mathematical Society, page 114,
By the property of the p'''-adic norm, (or by the “isosceles triangle principle”) we deduce that operatorname ordₚa_r=rλ₁.

2006, Matti Pitkanen, Topological Geometrodynamics, Luniver Press, page 531,
The definition of p-adic norm should obey the usual conditions, in particular the requirement that the norm of product is product of norms.

Given a prime p, we may define the p'''-adic norm #92;Vert#92;#92;Vert#95;p over the field of rational numbers #92;Q.

2012, Claire C. Ralph, Santiago R. Simanca, Arithmetic Differential Operators over the p-adic Integers, Cambridge University Press, page 2:

p-adic absolute value
p-adic absolute value

(number theory) A p-adic absolute value, for a given prime number p, the function, denoted |..|ₚ and defined on the rational numbers, such that |0|ₚ = 0 and, for x≠0, |x|ₚ = p^(-ordₚ(x)), where ordₚ(x) is the p-adic ordinal of x; the same function, extended to the p-adic numbers ℚₚ (the completion of the rational numbers with respect to the p-adic ultrametric defined by said absolute value); the same function, further extended to some extension of ℚₚ (for example, its algebraic closure).


p-adic ultrametric

p-adic ultrametric (plural p-adic ultrametrics)

On p'''-adic ultrametrics in the genome / In previous section we demonstrated that codons and amino acids are elements of some p'''-adic ultrametric spaces.

2017 September 15, Branko Dragovich, Andrei Yu. Khrennikov, Nataša Ž. Mišić, “Ultrametrics in the genetic code and the genome”, in Applied Mathematics and Computation, volume 309, →DOI, pages 350–358:

(number theory) The ultrametric with prime number p as parameter defined as d_p(x,y)=|x-y|ₚ; i.e., such that the distance between two rational numbers is equal to the p-adic absolute value of the difference between those two numbers.


trivial absolute value

trivial absolute value (plural trivial absolute values)

The trivial absolute value gives rise to the discrete metric.

(number theory) An absolute value (or norm) for a field which is defined to be equal to one for any element of the field other than the field's zero, for which it is defined to be equal to zero.


residue class

(number theory) A coset of an ideal of the ring of integers.


LTE

LTE (countable and uncountable, plural LTEs)

lifting the exponent
lifting the exponent#English

(number theory) Initialism of lifting the exponent. (a lemma used to calculate p-adic orders)


primitive root

There are #92;phi(p-1) incongruent primitive roots of p. The fact that there are so many primitive roots causes no difficulty in the theory of the binomial congruence but has caused considerable confusion in the tabulation of primitive roots.

1941, Derrick Henry Lehmer, Guide to Tables in the Theory of Numbers, National Research Council, page 13:

The integers 2, 3, 4, and 6 each have exactly one primitive root and therefore, by default, each has a set of primitive roots consisting of "consecutive" integers.
The integer 5, with primitive roots of 2 and 3 is the only positive integer having at least two primitive roots for which the entire set of primitive roots are consecutive integers.

1992, Joe Roberts, Lure of the Integers, Mathematical Association of America, page 55:

For example, the prime 7 has #92;phi(6)#61;2 primitive roots, namely, 3 and 5. Also, the prime 11 has #92;phi(10)#61;4 primitive roots, namely, 2, 6, 7, 8.
Recall from Theorem 6.7 that if m has primitive roots, and if g is one primitive root (#92;operatorname#123;mod#125;m), then we can obtain all primitive roots (#92;operatorname#123;mod#125;m) by raising g to appropriate exponents.

2006, Neville Robbins, Beginning Number Theory, Jones & Bartlett Learning, page 159:

(mathematics, number theory) For a given modulus n, a number g such that for every a coprime to n there exists an integer k such that gᵏ ≡ a (mod n); a generator (or primitive element) of the multiplicative group, modulo n, of integers relatively prime to n.

原根

number such that gk ≡ a (mod n) exists for every a coprime to n — see also generator, primitive element

generator

number that generates other numbers modulo n


cyclotomic field

cyclotomic field (plural cyclotomic fields)

A cyclotomic field is the splitting field of the cyclotomic polynomial #92;Phi#95;n(x)#61;#92;prod#95;#92;stackrel#123;1#92;lek#92;len#125;#123;#92;gcd(k,n)#61;1#125;#92;left(x-e#123;#92;frac#123;2k#92;pii#125;#123;n) and consequently is a Galois extension of the field of rational numbers.}}

Cyclotomic fields are fields obtained by adjoining to #92;Q roots of unity, i.e. roots of polynomials of the form Xⁿ-1, although the reader is warned that this terminology will be extended in §2.[…]Cyclotomic fields play a fundamental role in a number of arithmetic problems: for instance primes in arithmetic progression (see VIII,§4) and Fermat's Last Theorem (VII,§1).

1991, A. Fröhlich, M. J. Taylor, “Algebraic Number Theory”, in Paperback, Cambridge University Press, published 1993, page 205:

At about the same time Kummer introduced his "ideal numbers," defined an equivalence relation on them, and derived, for cyclotomic fields, certain special properties of the number of equivalence classes, the so-called class number of a cyclotomic field—in our terminology, the order of the ideal class group of the cyclotomic field.

2007, Israel Kleiner, A History of Abstract Algebra, Springer (Birkhäuser), page 26:

As in the case of the Drinfel'd double,¹³¹ it can be shown that these fields are subfields of the cyclotomic field #92;Q#95;N. Since the Galois group of #92;Q#95;N is abelian, every subfield of the cyclotomic field is normal, and consequently preserved by the action of the Galois group of #92;Q#95;N.

2012, Yorck Sommerhäuser, Yongchang Zhu, Hopf Algebras and Congruence Subgroups, American Mathematical Society, page 94:

(number theory, algebraic number theory) A number field obtained by adjoining a primitive root of unity to the field of rational numbers.

Kreisteilungskörper

extension field of the rational numbers formed by adjoining a primitive root of unity