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

wone (alternative obsolete)
I (alternative)

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

PIE word
 *wréh₂ds
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

radicale

ring theory: intersection of maximal submodules of a module

(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


elementary

en-adj

elementary (comparative more elementary, superlative most elementary)

more elementary (comparative)
most elementary (superlative)
elementar (alternative obsolete)
elem (alternative Philippines)

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

As magical square by 1670 and magic square by 1701 (see Citations). Calque of French quarré magique or Latin quadratum magicum. Magic refers to its original function as a talisman.

(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


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.

law of quadratic reciprocity (alternative)

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)

Pell's equations (plural)
Pell equation (alternative)

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)

Euclidean algorithms (plural)
Euclid's algorithm (alternative)

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”

LL-Q1860 (eng)-Pvanp7-greatest common divisor.wav

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)

Diophantine equations (plural)
diophantine equation (alternative)

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


continued fraction

continued fraction (plural continued fractions)

continued fractions (plural)
continued-fraction (alternative)

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.


Riemann zeta function

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

Riemann zeta functions (plural)
Riemann zeta-function (alternative)
Riemann's zeta function (alternative)

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


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


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.


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


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


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

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.


divisibility sequence

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.


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.


primorial

/pɹaɪ̯ˈmɔːɹi.əl/

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

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/√.


p-adic absolute value

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

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


Mersenne prime

Named after French theologian, philosopher, mathematician, and music theorist Marin Mersenne (1588–1648).

named-after

In addition to being the largest known Mersenne prime, this number is also currently the largest known prime number of any type.

2004, Sheldon Axler, “3: Mathematicians Versus the Silicon Age: Who Wins?”, in David F. Hayes, Tatiana Shubin, editors, Mathematical Adventures for Students and Amateurs, American Mathematical Society, page 20:

2005, Jean-Claude Bajard, Laurent Imbert, Thomas Plantard, Modular Number Systems: Beyond the Mersenne Family, Helena Handschuh, M. Anwar Hasan (editors), Selected Areas in Cryptography: 11th International Workshop, SAC 2004, Revised Selected Papers, Springer, LNCS 3357, page 159,
Mersenne numbers of the form 2ᵐ-1 are well known examples, but they are not useful for cryptography because there are only a few primes (the first Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, etc).

Mersenne primes, named after the French priest first describing them in the 17th century, are of the form of 2ᵖ-1, where the exponent p is a prime itself. Obviously 7 is a Mersenne prime, since 2³-1#61;7, and 3 is a prime. The interesting twist comes in the fact that there are certain primes that are double Mersenne primes. These are of the form 2#123;2ᵖ-1#125;-1, meaning that the exponent now is not just a prime, but it is itself a Mersenne prime. Since 2#123;2²-1#125;-1#61;7, 7 is the very first double Mersenne prime.

2013, Louis Komzsik, Magnificent Seven: The Happy Number, Trafford Publishing, page 71:

(number theory) A prime number which is one less than a power of two (i.e., is expressible in the form 2ⁿ-1; for example, 31=2⁵-1).


Fermat prime

Named after French lawyer and amateur mathematician Pierre de Fermat (1601–1665).

It is not hard to prove that the only way that the number 1#43;2ᵉ can possibly be prime is when e is a power of 2, and so all Fermat primes must have the form 1#43;2#123;2ᵃ#125; for integers a#92;ge 0. The numbers F#95;a#61;1#43;2#123;2ᵃ#125; are called Fermat numbers, and although it is true that every Fermat prime is a Fermat number, it is certainly not true that every Fermat number is prime.

2001, I. Martin Isaacs, Geometry for College Students, American Mathematical Society, page 201:

Rather surprisingly, it turns out that Fermat primes play an important special role in this context, a phenomenon hitherto unobserved in the arithmetic theory of Hecke groups, and, as a byproduct of our investigation, several new characterizations of Fermat primes are obtained.

2004, T. W. Müller, “12: Parity patterns in Hecke groups and Fermat primes”, in Thomas Wolfgang Müller, editor, Groups: Topological, Combinatorial and Arithmetic Aspects, Cambridge University Press, page 327:

F₀ = 3, F₁ = 5, F₂ = 17, F₃ = 257, and F₄ = 65537.
The next 28 Fermat numbers, F₅ through F₃₂, are known to be composite.

2013, Dean Hathout, Wearing Gauss’s Jersey, Taylor & Francis (CRC Press), page xi, Believe it or not, so far only five Fermat primes are known

(number theory) A Fermat number that is prime.


porism

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

From Ancient Greek πόρισμα (pórisma, “a deduction from a previous demonstration”).

der

(geometry, number theory) A proposition affirming the possibility of finding such conditions as will render a certain determinate problem indeterminate or capable of innumerable solutions.


Euler's criterion

Named after Leonhard Euler (1707–1783), Swiss mathematician and physicist, who devised the criterion in 1748.

(number theory) A theorem which states that, given a positive integer a which is coprime to an odd prime number p, a is a quadratic residue of p if and only if a^((p-1)/2) is congruent to 1 modulo p.


eigencurve

(mathematics, number theory) A rigid analytic curve that parametrizes certain p-adic families of modular forms.


eigenvariety

(mathematics, number theory) A higher-dimensional generalization of an eigencurve.


Bernoulli number

en-noun

Bernoulli number (plural Bernoulli numbers)

Bernoulli numbers (plural)
Bernoulli's number (alternative)

Named after Swiss mathematician Jacob Bernoulli (1654–1705), who discovered the numbers independently of and at about the same time as Japanese mathematician Seki Kōwa.
The numbers first appeared as coefficients in Bernoulli's formulae for the sum of the first n positive integers, each raised to a given power. The sequence was subsequently found in numerous other contexts.

1993, Serge Lang, Complex Analysis, Springer, 3rd Edition, page 418,
The assertion about the value of the zeta function at negative integers then comes immediately from the definition of the Bernoulli numbers in terms of the coefficients of a power series, namely
t/(eᵗ-1)=∑ₙ₌₀ ᪲B_n(tⁿ)/(n!)

In the first three sections, we present standard results about the Bernoulli numbers and polynomials and the Riemann zeta functions and its zeros.

2002, M. Aslam Chaudhry, Syed M. Zubair, On a Class of Incomplete Gamma Functions with Applications, CRC Press (Chapman & Hall/CRC), page 287:

This done, he extends occurrence of Bernoulli numbers in the expansion of #92;textstyle#92;frac 1 2#92;cot(#92;frac 1 2x) to the more general form #92;textstyle#92;frac#92;pin#92;cot(#92;frac#123;m#92;pi#125;n) and uses that to relate Bernoulli numbers to the values of #92;zeta(2n). To end the theoretical parts of his exposition, he gives some of the properties of the Bernoulli polynomials and notes that Bernoulli numbers grow faster than any geometric series.

2007, C. Edward Sandifer, How Euler Did It, Mathematical Association of America, page 175:

mathematical analysis
mathematical analysis

(mathematical analysis, number theory) Any one of the rational numbers in a sequence of such that appears in numerous contexts, including formulae for sums of integer powers and certain power series expansions.

Bernoulli number
Jacob Bernoulli
Seki Takakazu


abc conjecture

abc conjectures (plural)
abc-conjecture (alternative)
ABC conjecture (alternative)

From the names of the variables a, b, and c. The conjecture was proposed in 1985.

1985, Paul Vojta, Appendix, Serge Lang, Introduction to Arakelov Theory, Springer, 1988 Softcover, page 156,
Finally in §5 we give one application to the curve X⁴ + Y⁴ = Z⁴, showing that the height inequalities for the curve imply the asymptotic Fermat conjecture and a weak form of the Masser-Oesterlé abc''' conjecture.

The abc''' conjecture may well replace the Fermat theorem for the future generation of mathematicians.

2004, Sergei K. Lando, R.V. Gamkrelidze, V.A. Vassiliev, Graphs on Surfaces and Their Applications, Springer, page 137:

To prove or disprove the abc'''-conjecture would be an important contribution to number theory.[…]Langevin ([236], [237]) proved that the abc'''-conjecture implies the Erdős-Woods conjecture with k = 3 except perhaps a finite number of counter examples.

2006, Pei-Chu Hu, Chung-Chun Yang, Value Distribution Theory Related to Number Theory, Springer (Birkhäuser), page 233:

The abc'''-conjecture of Masser and Oesterle is a typical example of a simple statement that can be used to unify and motivate many results in number theory, which otherwise would be scattered statements without a common link.

2007, Enrico Bombieri, Walter Gubler, Heights in Diophantine Geometry, Cambridge University Press, page 401:

(number theory) Given coprime positive integers a, b and c, such that a + b = c, and d the radical of abc (the product of its distinct prime factors), the conjecture that d is usually not much smaller than c (in other words, that if a and b are divisible by large powers of primes, then c usually is not).


sociable number

sociable number (plural sociable numbers)

The numbers in the sequence <1264460, 1547860, 1727636, 1305184> are sociable numbers.

(number theory) Any of a sequence of numbers having the property that the sum of the divisors of each, excluding itself, is equal to the next number in the sequence; in the case of the last number of the sequence, the sum of the divisors, excluding that number, is equal to the first number of the sequence.

liczba towarzyska

number whose aliquot sums form a periodic sequence


Chen prime

Named after Chinese mathematician Chen Jingrun, who proved in 1966 that there are infinitely many such numbers.

2013, Laura Hemphill, Buying In, Haughton Mifflin Harcourt (New Harvest), page 278,
She tried counting Chen primes. 2, 3, 5... 7, 11... 13... 17... She couldn't concentrate.

There are even magic square of primes, like this one containing Chen primes discovered by Rudolf Ondrejka:[…]

2014, Christian Bessiere et al., “Reasoning about Constraint Methods”, in Duc-Nghia Pham, Seong-Bae Park, editors, PRICAI 2014: Trends in Artificial Intelligence, Springer, page 804:

A prime number p is called a Chen prime if p + 2 is either a prime or a product of two primes.[…]There are infinitely [many] Chen primes and the first ten are: 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.

2014, Rajesh Kumar Thakur, The Power of Mathematical Numbers, Ocean Books, page 155:

(number theory) Any prime number p such that p+2 is either a semiprime or another prime.


Farey sequence

Named after British geologist John Farey Sr., whose letter about the sequences was published in the Philosophical Magazine in 1816.

Students should then see the number of fractions N, in the Farey sequence is equal to #92;phi(2)#43;#92;phi(3)#43;#92;phi(4)#43;#92;dots#43;#92;phi(n), where #92;phi(n) is the number of positive integers less than or equal to n that are relatively prime to n.

2002, Alfred S. Posamentier, Jay Stepelman, Teaching Secondary Mathematics, Merrill, page 403:

2007, Jakub Pawlewicz, Order Statistics in the Farey Sequences in Sublinear Time, Lars Arge, Michael Hoffmann, Emo Welzl (editors), Algorithms - ESA 2007: 15th Annual European Symposium, Proceedings, Springer, LNCS 4698, page 218,
The Farey sequence of order n (denoted ℱₙ) is the increasing sequence of all irreducible fractions from interval [0,1] with denominators less than or equal to n. The Farey sequences have numerous interesting properties and they are well known in the number theory and in the combinatorics.

Riemann sums have also important connections with various problems from number theory, among them the Riemann Hypothesis, through their link with Farey sequences.

2009, Michel Weber, Dynamical Systems and Processes, European Mathematical Society, page 549:

(number theory) For a given positive integer n, the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size.

Farey sequence
John Farey Sr.
Philosophical Magazine


partition function

partition function (plural partition functions)

(number theory) function that represents the number of possible partitions of a natural number.


Dickman function

(number theory) A function, denoted by ρ, used to estimate the proportion of smooth numbers up to a given bound.


Liouville number

Liouville number (plural Liouville numbers)

Named after Joseph Liouville (1809–1882), a French mathematician.

(number theory) An irrational number x with the property that, for every positive integer n, there exist integers p and q with q > 1 and such that 0<|x-p/q|<1/(qⁿ).


Sierpinski number

Sierpinski number (plural Sierpinski numbers)

Named after Polish mathematician Wacław Sierpiński.

named-after

(number theory) An odd natural number k such that k×2ⁿ+1 is composite for all natural numbers n.


hyperperfect number

en-noun

hyperperfect number (plural hyperperfect numbers)

1966, American Mathematical Society Translations, page 258,
[…] the asymptotic density of all hyperperfect numbers, that is, numbers m for which m | σ(m), is equal to zero.

The rank of a hyperperfect number N is the ratio of divisor sum to N (which equals 2 for perfect numbers).

1974, William Judson LeVeque, editor, Reviews in number theory, as printed in Mathematical reviews, 1940 through 1972, volumes 1-44 inclusive, volume 1, page 107:

In 1974, Daniel Minoli and Robert Bear described a number of properties of hyperperfect numbers.

1999, James J. Tattersall, Elementary Number Theory in Nine Chapters, page 144:

(mathematics, number theory) Any natural number n for which, for some positive integer k, n = 1 + k(σ(n) - n - 1), where σ(n) is the sum of the positive divisors of n.


highly composite number

highly composite number (plural highly composite numbers)

Plato
Srinivasa Ramanujan
highly composite number

Coined by Indian mathematician Srinivasa Ramanujan in 1915, although it has been suggested that Plato may have known of the concept, since he specified 5040 (a highly composite number) as the ideal number of citizens in a city.

In the unpublished section of his notebook, Ramanujan extends the notion of highly composite number to other arithmetic functions, mainly to Q#95;#123;2k#125;(N),#92;1#92;lek#92;le 4, where Q#95;#123;2k#125;(N) denotes the number of representations of N as the sum of 2k squares, and to #92;sigma#95;#123;-s#125;(N), where #92;sigma#95;#123;-s#125;(N) denotes the sum of the (-s)th powers of the divisors of N.

2012, George E. Andrews, Bruce C. Berndt, Ramanujan's Lost Notebook, Part III, Springer, page 359:

Hardy has stated that a highly composite number is as unlike a prime as a number can be.

1998, K. Srinivasa Rao, Srinivasa Ramanujan: A Mathematical Genius, East West Books, page 48:

Ramanujan devoted a section of his paper to the study of Q(x), the number of highly composite numbers #92;lex Since d(2n)gt;d(n), we see that between x and 2x, there is always a highly composite number.

2013, M. Ram Murty, V. Kumar Murty, The Mathematical Legacy of Srinivasa Ramanujan, Springer, page 144:

A highly composite number, then, was in Hardy's phrase "as unlike a prime as a number can be." Ramanujan had explored their properties for some time; in the earliest pages of his second notebook he'd listed about a hundred highly composite numbers−the first few are 2, 4, 6, 12, 24, 36, 48, 60, 120−searching for patterns. He found them.

2013, Robert Kanigel, The Man Who Knew Infinity, Simon & Schuster, page 232:

(number theory) A positive integer that has more divisors than any smaller positive integer.

hochzusammengesetzte Zahl

positive integer with more divisors than any smaller positive integer


Goldbach's conjecture

“Goldbach’s conjecture, I finally remember it,” he says, as I’m standing to go. “And I want to make sure I tell you. It’s important: For every even number n>2, there exist two prime numbers p_1 and p_2 such that we can write: p_1 + p_2 = n…”

2022, Iain Reid, We Spread, Scribner, pages 255-256:

(number theory) A conjecture stating that every even integer greater than 2 can be expressed as the sum of two primes.


Lindemann-Weierstrass theorem

Named after Ferdinand von Lindemann and Karl Weierstrass, who worked on related proofs.

(number theory) A result that is useful in establishing the transcendence of numbers, stating that, if α₁, ..., αₙ are algebraic numbers which are linearly independent over the rational numbers ℚ, then e^(α₁), ..., e^(αₙ) are algebraically independent over ℚ.


hyperdeficient

en-adj

hyperdeficient (comparative more hyperdeficient, superlative most hyperdeficient)

more hyperdeficient (comparative)
most hyperdeficient (superlative)

(mathematics, number theory) Having the property that, for some positive integer k, δk(n) > 0, where δk(n) = n(k+1) +(k-1) –kσ(n) and σ(n) is the sum of the positive divisors of n.


anticyclotomic

Our methods are based on a new construction of the anticyclotomic p-adic L-function by means of the corresponding automorphic representation..

2015, Santiago Molina Blanco, “Anticyclotomic p-adic l-functions and the exceptional zero phenomenon”, in arXiv:

(number theory) Pertaining to a Galois character φ, such that φ . (φ ◦ c) = 1, where c is the nontrivial element of Gal(K/Q), where K is an imaginary quadratic field.


Mellin transform

en-noun

Mellin transform (plural Mellin transforms)

Mellin transforms (plural)
Mellin-transform (alternative)

Named after Finnish mathematician Hjalmar Mellin.

This chapter is concerned with the theory and applications of the Mellin transform. We derive the Mellin transform and its inverse from the complex Fourier transform. This is followed by several examples and the basic operational properties of Mellin transforms.

1995, Lokenath Debnath, Integral Transforms and Their Applications, CRC Press, page 211:

2005, Robb J. Muirhead, Aspects of Multivariate Statistical Theory, John Wiley & Sons, page 303,
If X is a positive random variable with density function f(x), the Mellin transform M(s) gives the (s-l)th moment of X. Hence Theorem 8.2.6 gives the Mellin transform of W evaluated at s=h+1; that is,
M(h+1)=E(Wʰ).
The inverse Mellin transform gives the density function of W.

In Chapters 2 and 7, the Mellin transform of the exponential function and the inverse Mellin transform of the Gamma function play key roles in demonstrating the equivalence of the modular relation and the functional equation. In proving the identities in this chapter, Mellin transforms also play central roles.

2008, Bruce C. Berndt, Marvin I. Knopp, Hecke's Theory of Modular Forms and Dirichlet Series, World Scientific, page 115:

(mathematical analysis, number theory, statistics) An integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform.


Skewes' number

Named after South African mathematician Stanley Skewes.

With this extension, the number system exceeds a number of previously-named giants, such as Archimedes' largest number, Kasner's googolplex, Skewes' number and Candelaria's googoltriplex.

1976, Word Ways: The Journal of Recreational Linguistics, volumes 9-10, A. Ross Eckler, page 178:

(number theory) Any of several extremely large numbers used as upper bounds for the smallest natural number x for which π(x)> operatorname li(x), where π is the prime-counting function and li is the logarithmic integral function. These bounds have since been improved by others.


emirp

The sequence of emirps begins: 13, 17, 31, 37, 71, 73, 79, 97, 107, ...

The first few emirps in base 12 (using the convention T for ten and E for eleven) are: 15, 51, 57, 5E, 75, E5, 107, ...

Therefore we do not yet know whether the general emirps go on forever or whether they too eventually end with a largest emirp of all.

1986, Malcolm E. Lines, A Number for your Thoughts, Taylor & Francis, page 39:

(number theory) A prime number that becomes a different prime when its decimal digits (or digits in some specified other base) are reversed.


Stark-Heegner theorem

Named after Kurt Heegner and Harold Stark, who contributed to the earliest proof.

(number theory) A theorem that states precisely which quadratic imaginary number fields admit unique factorization in their ring of integers.


Fermat's Last Theorem

Named after French lawyer and amateur mathematician Pierre de Fermat (1601–1665), who famously claimed to have a proof, although it was not successfully proven until 1994 by Andrew Wiles.

Another theorem, distinguished as Fermat's last Theorem, has obtained great celebrity on account of the numerous attempts that have been made to demonstrate it.

1872, William Thomas Brande, editor, A Dictionary of Science, Literature, & Art:

A lot has been written about Fermat's Last Theorem since its proof was announced in 1993.

2002, Peter Hilton, Derek Holton, Jean Pedersen, Mathematical Vistas: From a Room with Many Windows, Springer, page 23:

2002, Brendan Kelly, Algebra with the TI-83 Plus & TI-83 Plus SE, Brendan Kelly Publishing, page 36,
It appeared that Dr. Wiles had proved Fermat's Last Theorem, the most famous conjecture in Number Theory which had eluded the greatest mathematicians for over 350 years!

Then, almost casually, Dr. Wiles ended his lecture with these words: "And by the way, this means that Fermat's Last Theorem was true. Q.E.D."

2007, Eli Maor, The Pythagorean Theorem: A 4,000-year History, Princeton University Press, page 1:

(number theory) The theorem that the Diophantine equation aⁿ+bⁿ=cⁿ has no solutions for positive integers a,b,c,n, where n>2.

費馬大定理 /费马大定理

theorem that an + bn equal to cn has no positive integer solutions for a, b, c, n with n > 2


Fermat's little theorem

Named after French lawyer and amateur mathematician Pierre de Fermat (1601–1665), who stated a version of the theorem in a letter in 1640. Called little to distinguish it from Fermat's Last Theorem.

1999, John Stillwell, Translator's introduction, Peter Gustav Lejeune Dirichlet, Richard Dedekind (supplements), Lectures on Number Theory, [1863, P. G. Lejeune Dirichlet, R. Dedekind, Vorlesungen über Zahlentheorie], American Mathematical Society, page xi,
When combined with the historical remarks made by Gauss himself, they give a bird's eye view of number theory from approximately 1640 to 1840 - from Fermat's little theorem to L-functions - the period which produced the problems and ideas which are still at the center of the subject.

1999, Siguna Müller, On the Combined Fermat/Lucas Probable Prime Test, Michael Walker (editor), Cryptography and Coding: 7th IMA International Conference, Springer, LNCS 1746, page 222,
Most of the pseudoprimality tests originate in some sense on Fermat's Little Theorem aⁿ⁻¹ ≡ 1 mod n.

Incidentally, the special case of Fermat's little theorem for a = 2 was known to the Chinese as early as 500 B.C.
The first proof of Fermat's little theorem was given by Euler in 1736, almost a century after Fermat's announcement.

2007, Thomas Koshy, Elementary Number Theory with Applications, 2nd edition, Elsevier (Academic Press), page 327:

Fermat's little theorem says that if n is prime and gcd(a,n) = 1, then aⁿ⁻¹ ≡ 1 (mod n), so it follows that n must be composite, even though we have not produced a factor.

2008, Lawrence C. Washington, Elliptic Curves: Number Theory and Cryptography, 2nd edition, Taylor & Francis (Chapman & Hall/CRC), page 189:

(number theory) The theorem that, for any prime number p and integer a, aᵖ-a is an integer multiple of p.


quadratic form

Quadratic form
Quadratic form (statistics)

Historically, quadratic form theory has been treated as a rich but misunderstood uncle.

1980, John Gregory, Quadratic Form Theory and Differential Equations, Academic Press, page ix:

2004, Nikita A. Karpenko, Izhboldin's Results on Stably Birational of Equivalence Quadrics, Oleg T. Izhboldin, Geometric Methods in the Algebraic Theory of Quadratic Forms, Springer, Lecture Notes in Mathematics 1835, page 156,
We claim that this quadratic form is isotropic, and this gives what we need according to [2, Proposition 4.4].

2009, R. Parimala, V. Suresh, J.-P. Tignol, On the Pfister Number of Quadratic Forms, Ricardo Baeza, Quadratic Forms—Algebra, Arithmetic, and Geometry, American Mathematical Society, page 327,
The generic quadratic form of even dimension n with trivial discriminant over an arbitrary field of characteristic different from 2 containing a square root of −1 can be written in the Witt ring as a sum of 2-fold Pfister forms using n − 2 terms and not less. The number of Pfister forms required to express a quadratic form of degree 6 with trivial discriminant is determined in various cases.

The arithmetic theory of binary quadratic forms, which had its origins in the work of Fermat, was extensively developed during the 18th century by Euler, Lagrange, Legendre and Gauss. The extension to quadratic forms in more than two variables, which was begun by them and is exemplified by Lagrange's theorem that every positive integer is a sum of four squares, was continued in the 19th century by Dirichlet, Hermite, H. J. S. Smith, Minkowski and others.

2009, W. A. Coppel, Number Theory: An Introduction to Mathematics, 2nd edition, Springer, page 291:

homogeneous polynomial
homogeneous polynomial

(mathematics, number theory, algebra) A homogeneous polynomial of degree 2 in a given number of variables.


quadratic residue

quadratic residue (plural quadratic residues)

quadratic nonresidue

antonym(s) of “integer expressible as a square modulo n”

There are many small tables of quadratic residues giving for the first few primes p the positive quadratic residues of p arranged in order of their size.

1941, Derrick Henry Lehmer, Guide to Tables in the Theory of Numbers, National Research Council, page 52:

1991, John Stillwell (translator), Peter Gustav Lejeune Dirichlet, Lectures on Number Theory, [1863, Vorlesungen über Zahlentheorie], American Mathematical Society, London Mathematical Society, page 82,
If we now make the assumption that q is a quadratic residue of all odd primes z not greater than 2m + 1, then it follows from earlier theorems (§37) that the prime q, since it is ≡ 1 (mod 8) and hence a quadratic residue of each power of 2, is also a quadratic residue of each number which has no odd prime factors except the prime numbers z

1999, Dinakar Ramakrishnan, Robert J. Valenza, Tsinghua University Press [清华大学出版社有限公司], Fourier Analysis on Number Fields, page 213,
Of special importance here is the quadratic reciprocity law, which for primes p and q gives a precise relationship between the status of p as a quadratic residue mod q and the status of q as a quadratic residue mod p.

(number theory, modular arithmetic) For given positive integer n, any integer that is congruent to some square m² modulo n.

kvadratisk rest

integer that is congruent to the square of some other number, to a given modulus


modular arithmetic

modular arithmetic (countable and uncountable, plural modular arithmetics)

The reader now has examined, in some detail, several specific modular arithmetics, namely, #92;mathbb#123;Z#125;#95;2,#92;mathbb#123;Z#125;#95;5,#92;mathbb#123;Z#125;#95;6,#92;mathbb#123;Z#125;#95;7,#92;mathbb#123;Z#125;#95;8 and #92;mathbb#123;Z#125;#95;9.

1969, Joseph Landin, An Introduction to Algebraic Structures, Dover, page 153:

In this section we use two historical ciphers to introduce modular arithmetic with integers. Even though the historical ciphers are no longer relevant, modular arithmetic is extremely important in modern cryptography, especially for asymmetric algorithms.

2010, Christof Paar, Jan Pelzl, Understanding Cryptography: A Textbook for Students and Practitioners, Springer, page 13:

Hence for prime moduli, modular arithmetic is very similar to regular rational arithmetic with all four operations defined.

1997, Robert E. Jamison, “Rhythm and Pattern: Discrete Mathematics with an Artistic Connection for Elementary School Teachers”, in Joseph G. Rosenstein, Deborah S. Franzblau, Fred S. Roberts, editors, Discrete Mathematics in the Schools, American Mathematical Society, page 215:

(number theory) Any system of arithmetic for integers which, for some given positive integer n, is equivalent to the set of integers being mapped onto the finite set {0, ... n} according to congruence modulo n, and in which addition and multiplication are defined consistently with the results of ordinary arithmetic being so mapped.


ideal number

Apparently a calque of German ideale Zahl, a concept in number theory developed by German mathematician Ernst Kummer (1810—1893) and later incorporated by Richard Dedekind into the ring theory concept of ideal.

1861, H. J. Stephen Smith, Report on the Theory of Numbers—Part II, Report of the Thirtieth Meeting of the British Association for the Advancement of Science, June-July 1860, British Association for the Advancement of Science, page 133,
This symbolic representation of ideal numbers is very convenient, and tends to abbreviate many demonstrations.
Every ideal number is a divisor of an actual number, and, indeed, of an infinite number of actual numbers. Also, if the ideal number ϕ(α) be a divisor of the actual number F(α), the quotient ϕ₁(α)=F(α)÷ϕ(α) is always ideal; for if ϕ₁(α) were an actual number, ϕ(α), which is the quotient of F(α) divided by ϕ₁(α), ought also to be an actual number.

1992 [World Scientific], C. Y. Hsiung, Elementary Theory of Numbers, 1995, Allied Publishers, page 214,
One of the most important developments of algebraic number theory is the creation of ideal numbers. All ideal numbers can be classified into ideal number classes. Both the study of ideal numbers and the computation of ideal number classes are important problems of algebraic number theory. The introduction of ideal numbers was meant originally for the number theory, but now ideal numbers have become an important tool in algebra.

1995, L. V. Kuz'min, Ideal number, entry in M. Hazewinkel (editor) Encyclopaedia of Mathematics: Volume 3, Springer, page 125,
The semi-group D is a free commutative semi-group with identity; its free generators are called prime ideal numbers. In modern terminology, ideal numbers are known as integral divisors of A.

(number theory) An algebraic integer that represents an ideal in the ring of integers of a number field.

Ernst Kummer
Richard Dedekind
ideal number

ideale Zahl

algebraic number that represents an ideal in the ring of integers


Riemann hypothesis

Named after German mathematician Bernhard Riemann (1826–1866), who first formulated and discussed the hypothesis.

The Riemann hypothesis has deep implications about the distribution of prime numbers.

1995, John Corning Carey, On Beurling's Approach to the Reimann Hypothesis, University of California, Berkeley, page 43,
But in the absence of such assumptions, the task of finding functions h∈ℱ₂ for which ‖1+h‖₂ is small is equivalent to proving the Riemann hypothesis, as we will now demonstrate.

2003, Marcus du Sautoy, The Music of the Primes, 2004, HarperCollins Publishers (Harper Perennial), page 10,
A solution of the Riemann Hypothesis will have huge implications for many other mathematical problems.

2010, Samuel J. Patterson, The Riemann Hypothesis – a short history, Gerrit Dijk, Masato Wakayama (editors), Casimir Force, Casimir Operators and the Riemann Hypothesis, Walter de Gruyter, page 30,
The one problem proposed in Riemann's paper which remained unproved, the only one Riemann put forward explicitly as a conjecture, was the Riemann Hypothesis.

2021, Naji Arwashan, The Riemann Hypothesis and the Distribution of Prime Numbers, Nova Science Publishers, page x,
The Riemann Hypothesis is considered by many accounts the single most important and difficult question in math today.

Riemann zeta function
Riemann zeta function

(mathematics, mathematical analysis, number theory) The conjecture that the zeros of the Riemann zeta function exist only at the negative even integers and certain complex numbers whose real part is ½.

hypothèse de Riemann

conjecture about the zeros of the Riemann zeta function


Brun's constant

(number theory) The finite value, denoted by B₂, to which the sum of the reciprocals of the twin primes converge, according to Brun's theorem.


Brun's theorem

(number theory) A theorem stating that the sum of the reciprocals of the twin primes converges to a finite value (Brun's constant).


Størmer's theorem

(number theory) A theorem that gives a finite bound on the number of consecutive pairs of smooth numbers that exist, for a given degree of smoothness, and provides a method for finding all such pairs using Pell equations.


p-adic

From p + -adic. The letter p follows common number theory usage in representing an arbitrary prime number; the suffix -adic signals that properties of the number p (generally, those due to its being prime, and specifically those of a specified p) are fundamental to the theory and determine the properties of the resulting construction.

The theory of p'''-adic analytic functions in domains other than simple disks is not very well known yet, although such kind of functions happens to intervene in questions linked to p'''-adic functional analysis, number theory, and others.

1995, Alain Escassut, Analytic Elements in P-adic Analysis, World Scientific, page v:

1997, Fernando Q. Gouvêa, p-adic Numbers: An Introduction, Springer, 2nd Edition, page 60,
Definition 3.3.3 The ring of p'''-adic integers is the valuation ring
Z ₚ=x∈ Q ₚ:|x|ₚ<1.

2. Preliminaries
3. A p'''-adic Cauchy problem
4. The p'''-adic Hilbert transform
5. The boundaries in the p'''-adic space L^q
[…]

2007, Nguyen Minh Cheung, Nguyen Van Co, Le Quang Thuan, §12: Harmonic Analysis Over P-adic Field I: Some Equations and Singular Field Operators, N. M. Chuong, Yu V. Egorov, A. Khrennikov, Y. Meyer, D. Mumford (editors), Harmonic, Wavelet and P-adic Analysis, World Scientific, page 272, The paper is organized as follows

(number theory) Of, pertaining to, (ultimately) derived from or defined in the context of p-adic numbers.


prime number theorem

prime number theorem (plural prime number theorems)

But we cannot infer from them the equivalence in any sense of these two propositions, since we have used in our proof of the prime number theorem a subsidiary theorem on the order of magnitude of #92;zeta'(s)#47;#92;zeta(s).

1932, A. E. Ingham, Albert Edward Ingham, The Distribution of Prime Numbers, Cambridge University Press, page 39:

1974 [Academic Press], Harold M. Edwards, Riemann's Zeta Function, 2001, Dover, page 182,
The problem of locating the roots p of ζ, and consequently the problem of estimating the error in the prime number theorem, is closely related to the problem of estimating the growth of ζ in the critical strip 0<Res<1 as Ims→∞.

In 1859, Riemann attempted to give a complete proof of the prime number theorem using the zeta function for complex variables s. Although he was not successful in proving the prime number theorem he established many properties of the zeta function and showed that the prime number theorem depended on the zeros of the zeta function.

2016, Benjamin Fine, Gerhard Rosenberger, Number Theory: An Introduction via the Density of Primes, 2nd edition, Springer (Birkhäuser), page 145:

(number theory) The theorem that the number of prime numbers less than n asymptotically approaches n / ln(n) as n approaches infinity.

In [6], [11], abstract prime number theorems are proved under a variety of conditions.

1996, Illinois Journal of Mathematics, volume 40, University of Illinois Press, page 245:

(number theory) Any theorem that concerns the distribution of prime numbers.


superabundant number

superabundant number (plural superabundant numbers)

1984, Richard K. Guy (editor), Reviews in Number Theory 1973-83, Volume 4, Part 1, As printed in Mathematical Reviews, American Mathematical Society, page 173,
The authors prove a theorem: If Q(X) is the number of superabundant numbers ≦X, then for c<5/48,Q(X)≧( log X)¹⁺ᶜ for sufficiently large X.

1995, József Sándor, Dragoslav S. Mitrinović, Borislav Crstici, Handbook of Number Theory I, Springer, page 111,
1) A number n is called superabundant if (σ(n))/n>(σ(m))/m for all m with 1Corollary. Q(x)>c log x log log x/( log log log x)².

It follows that there is a superabundant number in each real interval #91;x,2x#93; and that the number of superabundant numbers is thus infinite.

2017, Kevin Broughan, Equivalents of the Riemann Hypothesis: Volume 1, Arithmetic Equivalents, Cambridge University Press, page 151:

(number theory) A positive integer whose abundancy index is greater than that of any lesser positive integer.


Dirichlet series

Dirichlet series (countable and uncountable, plural Dirichlet series)

Dirichlet series (plural)
Dirichlet's series (alternative)

Named after German mathematician Peter Gustav Lejeune Dirichlet (1805–1859).

named-after

Traditionally, starting from Euler, multiplicativity of arithmetic sequences is customarily expressed in the form of an Euler product factorization of the generating Dirichlet series. It turns out that in the situation of modular forms, suitable Dirichlet series constructed by Fourier coefficients of eigenfunctions of Hecke operators can be expressed through Dirichlet series formed by the corresponding eigenvalues.

2009, Anatoli Andrianov, Introduction to Siegel Modular Forms and Dirichlet Series, Springer (Birkhäuser), page 137:

We have now given heuristically a large family of multiple Dirichlet series, one for each simply laced Dynkin diagram.

2012, Daniel Bump, “Chapter 1: Introduction: Multiple Dirichlet Series”, in Daniel Bump, Solomon Friedberg, Dorian Goldfeld, editors, Multiple Dirichlet Series, L-functions and Automorphic Forms, Springer, page 6:

2014, Marius Overholt, A Course in Analytic Number Theory, American Mathematical Society, page 157,
The sum
A(s)=∑ₙ₌₁ ᪲a_nn⁻ˢ
of a convergent Dirichlet series is a holomorphic (single-valued analytic) function in the half plane σ>σ_c(A), and the terms of the Dirichlet series are holomorphic in the whole complex plane, and the series converges uniformly on every compact subset of σ>σ_c(A) by Proposition 3.3.

(number theory) Any infinite series of the form ∑ₙ₌₁ ᪲(a_n)/(nˢ), where s and each a_n are complex numbers.