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)

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


(a) (rare) egressive airstream mechanism when used with a letter for an ingressive sound, e.g. [ʘ↑]

(b) exhalation when used alone, [↑].

(number theory, computer science) exponentiation


abc conjecture

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.


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


質數

/ʈ͡ʂʐ̩⁵¹⁻⁵³ ʂu⁵¹/

/ʈ͡ʂʐ̩³⁵ ʂu⁵¹/

/ʈ͡ʂʐ̩⁵¹⁻⁵³ ʂu⁵¹/

/ʈ͡ʂʐ̩³⁵ ʂu⁵¹/


partition function

partition function (plural partition functions)

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


素數

/su⁵¹⁻⁵³ ʂu⁵¹/

/sɔ²¹⁻⁵³ sɔ²¹/

/sɔ⁴¹⁻⁵⁵⁴ sɔ⁴¹/

/sɔ²¹⁻⁵³ sɔ²¹/

/sɔ¹¹⁻⁵³ sɔ¹¹/

/sɔ²¹⁻⁴¹ sɔ²¹/

/su⁵¹⁻⁵³ ʂu⁵¹/

/sɔ²¹⁻⁵³ sɔ²¹/

/sɔ⁴¹⁻⁵⁵⁴ sɔ⁴¹/

/sɔ²¹⁻⁵³ sɔ²¹/

/sɔ¹¹⁻⁵³ sɔ¹¹/

/sɔ²¹⁻⁴¹ sɔ²¹/


número amigo

(number theory) amicable number (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)


número amigável

pt-noun

número amigável m (plural números amigáveis)


Dickman function

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


fachtóir príomha

head

fachtóir príomha m (genitive singular fachtóra phríomha, nominative plural fachtóirí príomha)

fachtóra phríomha (genitive singular)
fachtóirí príomha (nominative plural)
nominative (indefinite)
fachtóir príomha (indefinite singular)
fachtóirí príomha (indefinite plural)
vocative (indefinite)
a fhachtóir phríomha (indefinite singular)
a fhachtóirí príomha (indefinite plural)
genitive (indefinite)
fachtóra phríomha (indefinite singular)
dative (indefinite)
nominative (definite)
an fachtóir príomha (definite singular)
na fachtóirí príomha (definite plural)
genitive (definite)
an fhachtóra phríomha (definite singular)
na bhfachtóirí príomha (definite plural)
dative (definite)
leis an bhfachtóir príomha (definite singular)
don fhachtóir príomha (definite singular)
leis na fachtóirí príomha (definite plural)
fachtóir príomha
fhachtóir príomha
bhfachtóir príomha


Liouville number

en-noun

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).


正數

/ʈ͡ʂɤŋ⁵¹⁻⁵³ ʂu⁵¹/

/t͡sɪŋ³³ sou̯³³/

/ʈ͡ʂɤŋ⁵¹⁻⁵³ ʂu⁵¹/

/t͡sɪŋ³³ sou̯³³/


最大公約數

ㄗㄨㄟˋ ㄉㄚˋ ㄍㄨㄥ ㄩㄝ ㄕㄨˋ

цзуйда гунъюэшу

/t͡su̯eɪ̯⁵¹⁻⁵³ tä⁵¹ kʊŋ⁵⁵ ɥɛ⁵⁵ ʂu⁵¹/

/t͡sɵy̯³³ taːi̯²² kʊŋ⁵⁵ jœːk̚³ sou̯³³/

/t͡su̯eɪ̯⁵¹⁻⁵³ tä⁵¹ kʊŋ⁵⁵ ɥɛ⁵⁵ ʂu⁵¹/

/t͡sɵy̯³³ taːi̯²² kʊŋ⁵⁵ jœːk̚³ sou̯³³/

The greatest common divisor of 12 and 10 is 2.

12和10的最大公約數是2。

12和10的最大公约数是2。

greatest common divisor
greatest common divisor

greatest common factor
greatest common factor

highest common factor
highest common factor

(arithmetic, number theory) greatest common divisor; greatest common factor; highest common factor


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.


最小公倍數

ㄗㄨㄟˋ ㄒㄧㄠˇ ㄍㄨㄥ ㄅㄟˋ ㄕㄨˋ

цзуйсяо гунбэйшу

/t͡su̯eɪ̯⁵¹ ɕi̯ɑʊ̯²¹⁴⁻²¹ kʊŋ⁵⁵ peɪ̯⁵¹⁻⁵³ ʂu⁵¹/

/t͡sɵy̯³³ siːu̯³⁵ kʊŋ⁵⁵ pʰuːi̯¹³ sou̯³³/

The least common multiple of 12 and 10 is 60.

12和10的最小公倍數是60。

12和10的最小公倍数是60。

least common multiple
least common multiple

(arithmetic, number theory) least common multiple


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.


合成數

/xɤ³⁵ ʈ͡ʂʰɤŋ³⁵ ʂu⁵¹/

ㄏㄜˊ ㄔㄥˊ ㄕㄨˋㄦ

/xɤ³⁵ ʈ͡ʂʰɤŋ³⁵ ʂuɻʷ⁵¹/

/hɐp̚² sɪŋ²¹ sou̯³³/

/xɤ³⁵ ʈ͡ʂʰɤŋ³⁵ ʂu⁵¹/

/xɤ³⁵ ʈ͡ʂʰɤŋ³⁵ ʂuɻʷ⁵¹/

/hɐp̚² sɪŋ²¹ sou̯³³/

(number theory) Synonym of 合數 /合数 (héshù, “composite number”)


合數

/xɤ³⁵ ʂuɻʷ⁵¹/

/xɤ³⁵ ʂuɻʷ⁵¹/


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.


sfd

The gcd of 20 and 15 is 5.

(number theory) gcd (greatest common divisor)


mfm

Abbreviation of mindste fælles multiplum.

abbreviation of

The lcm of 6 and 8 is 24.

(number theory) lcm (least common multiple)


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)

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.

(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.


哥德巴赫猜想

ㄍㄜ ㄉㄜˊ ㄅㄚ ㄏㄜˋ ㄘㄞ ㄒㄧㄤˇ

Гэдэбахэ цайсян

/kɤ⁵⁵ tɤ³⁵ pä⁵⁵ xɤ⁵¹ t͡sʰaɪ̯⁵⁵ ɕi̯ɑŋ²¹⁴⁻²¹⁽⁴⁾/

/kɔː⁵⁵ tɐk̚⁵ paː⁵⁵ hɐk̚⁵ t͡sʰaːi̯⁵⁵ sœːŋ³⁵/

Goldbach's conjecture
Goldbach's conjecture


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.


kongruenssi

kongruenssi (nominative singular)
kongruenssit (nominative plural)
kongruenssi (accusative nominative singular)
kongruenssit (accusative nominative plural)
kongruenssin (accusative genitive singular)
kongruenssit (accusative genitive plural)
kongruenssin (genitive singular)
kongruenssien (genitive plural)
kongruenssia (partitive singular)
kongruensseja (partitive plural)
kongruenssissa (inessive singular)
kongruensseissa (inessive plural)
kongruenssista (elative singular)
kongruensseista (elative plural)
kongruenssiin (illative singular)
kongruensseihin (illative plural)
kongruenssilla (adessive singular)
kongruensseilla (adessive plural)
kongruenssilta (ablative singular)
kongruensseilta (ablative plural)
kongruenssille (allative singular)
kongruensseille (allative plural)
kongruenssina (essive singular)
kongruensseina (essive plural)
kongruenssiksi (singular translative)
kongruensseiksi (plural translative)
kongruenssitta (abessive singular)
kongruensseitta (abessive plural)
kongruenssein (instructive plural)
See the possessive forms below. (comitative)
kongruenssini (nominative singular)
kongruenssini (nominative plural)
kongruenssini (accusative nominative singular)
kongruenssini (accusative nominative plural)
kongruenssini (accusative genitive singular)
kongruenssini (accusative genitive plural)
kongruenssini (genitive singular)
kongruenssieni (genitive plural)
kongruenssiani (partitive singular)
kongruenssejani (partitive plural)
kongruenssissani (inessive singular)
kongruensseissani (inessive plural)
kongruenssistani (elative singular)
kongruensseistani (elative plural)
kongruenssiini (illative singular)
kongruensseihini (illative plural)
kongruenssillani (adessive singular)
kongruensseillani (adessive plural)
kongruenssiltani (ablative singular)
kongruensseiltani (ablative plural)
kongruenssilleni (allative singular)
kongruensseilleni (allative plural)
kongruenssinani (essive singular)
kongruensseinani (essive plural)
kongruenssikseni (singular translative)
kongruensseikseni (plural translative)
kongruenssittani (abessive singular)
kongruensseittani (abessive plural)
kongruensseineni (comitative plural)
kongruenssisi (nominative singular)
kongruenssisi (nominative plural)
kongruenssisi (accusative nominative singular)
kongruenssisi (accusative nominative plural)
kongruenssisi (accusative genitive singular)
kongruenssisi (accusative genitive plural)
kongruenssisi (genitive singular)
kongruenssiesi (genitive plural)
kongruenssiasi (partitive singular)
kongruenssejasi (partitive plural)
kongruenssissasi (inessive singular)
kongruensseissasi (inessive plural)
kongruenssistasi (elative singular)
kongruensseistasi (elative plural)
kongruenssiisi (illative singular)
kongruensseihisi (illative plural)
kongruenssillasi (adessive singular)
kongruensseillasi (adessive plural)
kongruenssiltasi (ablative singular)
kongruensseiltasi (ablative plural)
kongruenssillesi (allative singular)
kongruensseillesi (allative plural)
kongruenssinasi (essive singular)
kongruensseinasi (essive plural)
kongruenssiksesi (singular translative)
kongruensseiksesi (plural translative)
kongruenssittasi (abessive singular)
kongruensseittasi (abessive plural)
kongruensseinesi (comitative plural)
kongruenssimme (nominative singular)
kongruenssimme (nominative plural)
kongruenssimme (accusative nominative singular)
kongruenssimme (accusative nominative plural)
kongruenssimme (accusative genitive singular)
kongruenssimme (accusative genitive plural)
kongruenssimme (genitive singular)
kongruenssiemme (genitive plural)
kongruenssiamme (partitive singular)
kongruenssejamme (partitive plural)
kongruenssissamme (inessive singular)
kongruensseissamme (inessive plural)
kongruenssistamme (elative singular)
kongruensseistamme (elative plural)
kongruenssiimme (illative singular)
kongruensseihimme (illative plural)
kongruenssillamme (adessive singular)
kongruensseillamme (adessive plural)
kongruenssiltamme (ablative singular)
kongruensseiltamme (ablative plural)
kongruenssillemme (allative singular)
kongruensseillemme (allative plural)
kongruenssinamme (essive singular)
kongruensseinamme (essive plural)
kongruenssiksemme (singular translative)
kongruensseiksemme (plural translative)
kongruenssittamme (abessive singular)
kongruensseittamme (abessive plural)
kongruensseinemme (comitative plural)
kongruenssinne (nominative singular)
kongruenssinne (nominative plural)
kongruenssinne (accusative nominative singular)
kongruenssinne (accusative nominative plural)
kongruenssinne (accusative genitive singular)
kongruenssinne (accusative genitive plural)
kongruenssinne (genitive singular)
kongruenssienne (genitive plural)
kongruenssianne (partitive singular)
kongruenssejanne (partitive plural)
kongruenssissanne (inessive singular)
kongruensseissanne (inessive plural)
kongruenssistanne (elative singular)
kongruensseistanne (elative plural)
kongruenssiinne (illative singular)
kongruensseihinne (illative plural)
kongruenssillanne (adessive singular)
kongruensseillanne (adessive plural)
kongruenssiltanne (ablative singular)
kongruensseiltanne (ablative plural)
kongruenssillenne (allative singular)
kongruensseillenne (allative plural)
kongruenssinanne (essive singular)
kongruensseinanne (essive plural)
kongruenssiksenne (singular translative)
kongruensseiksenne (plural translative)
kongruenssittanne (abessive singular)
kongruensseittanne (abessive plural)
kongruensseinenne (comitative plural)
kongruenssinsa (nominative singular)
kongruenssinsa (nominative plural)
kongruenssinsa (accusative nominative singular)
kongruenssinsa (accusative nominative plural)
kongruenssinsa (accusative genitive singular)
kongruenssinsa (accusative genitive plural)
kongruenssinsa (genitive singular)
kongruenssiensa (genitive plural)
kongruenssiaan (partitive singular)
kongruenssiansa (partitive singular)
kongruenssejaan (partitive plural)
kongruenssejansa (partitive plural)
kongruenssissaan (inessive singular)
kongruenssissansa (inessive singular)
kongruensseissaan (inessive plural)
kongruensseissansa (inessive plural)
kongruenssistaan (elative singular)
kongruenssistansa (elative singular)
kongruensseistaan (elative plural)
kongruensseistansa (elative plural)
kongruenssiinsa (illative singular)
kongruensseihinsa (illative plural)
kongruenssillaan (adessive singular)
kongruenssillansa (adessive singular)
kongruensseillaan (adessive plural)
kongruensseillansa (adessive plural)
kongruenssiltaan (ablative singular)
kongruenssiltansa (ablative singular)
kongruensseiltaan (ablative plural)
kongruensseiltansa (ablative plural)
kongruenssilleen (allative singular)
kongruenssillensa (allative singular)
kongruensseilleen (allative plural)
kongruensseillensa (allative plural)
kongruenssinaan (essive singular)
kongruenssinansa (essive singular)
kongruensseinaan (essive plural)
kongruensseinansa (essive plural)
kongruenssikseen (singular translative)
kongruenssiksensa (singular translative)
kongruensseikseen (plural translative)
kongruensseiksensa (plural translative)
kongruenssittaan (abessive singular)
kongruenssittansa (abessive singular)
kongruensseittaan (abessive plural)
kongruensseittansa (abessive plural)
kongruensseineen (comitative plural)
kongruensseinensa (comitative plural)

From Latin congruentia via Germanic languages, compare German Kongruenz, Swedish kongruens.


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:

Diophantine equation
Diophantine equation

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

dernier théorème de Fermat

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.


kelipatan persekutuan terkecil

id-noun

kelipatan persekutuan terkecil (uncountable)


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


prímtala

is-noun

prímtala f (genitive singular prímtölu, nominative plural prímtölur)

prímtölu (genitive singular)
prímtölur (nominative plural)
prímtala (indefinite nominative singular)
prímtalan (definite nominative singular)
prímtölur (indefinite nominative plural)
prímtölurnar (definite nominative plural)
prímtölu (accusative indefinite singular)
prímtöluna (accusative definite singular)
prímtölur (accusative indefinite plural)
prímtölurnar (accusative definite plural)
prímtölu (dative indefinite singular)
prímtölunni (dative definite singular)
prímtölum (dative indefinite plural)
prímtölunum (dative definite plural)
prímtölu (genitive indefinite singular)
prímtölunnar (definite genitive singular)
prímtalna (genitive indefinite plural)
prímtalnanna (definite genitive plural)


最大公因數

ㄗㄨㄟˋ ㄉㄚˋ ㄍㄨㄥ ㄧㄣ ㄕㄨˋ

цзуйда гунъиньшу

/t͡su̯eɪ̯⁵¹⁻⁵³ tä⁵¹ kʊŋ⁵⁵ in⁵⁵ ʂu⁵¹/

/t͡sɵy̯³³ taːi̯²² kʊŋ⁵⁵ jɐn⁵⁵ sou̯³³/

/t͡su̯eɪ̯⁵¹⁻⁵³ tä⁵¹ kʊŋ⁵⁵ in⁵⁵ ʂu⁵¹/

/t͡sɵy̯³³ taːi̯²² kʊŋ⁵⁵ jɐn⁵⁵ sou̯³³/

(arithmetic, number theory) greatest common divisor; greatest common factor; highest common factor


最大公因子

ㄗㄨㄟˋ ㄉㄚˋ ㄍㄨㄥ ㄧㄣ ㄗˇ

цзуйда гунъиньцзы

/t͡su̯eɪ̯⁵¹⁻⁵³ tä⁵¹ kʊŋ⁵⁵ in⁵⁵ t͡sz̩²¹⁴⁻²¹⁽⁴⁾/

/t͡sɵy̯³³ taːi̯²² kʊŋ⁵⁵ jɐn⁵⁵ t͡siː³⁵/

/t͡sɵy̯³³ taːi̯²² kʊŋ⁵⁵ jɐn⁵⁵ t͡siː³⁵/


同餘

/tʰʊŋ³⁵ y³⁵/

/tʰʊŋ²¹ jyː²¹/

/tʰʊŋ³⁵ y³⁵/

/tʰʊŋ²¹ jyː²¹/


費馬小定理

ㄈㄟˋ ㄇㄚˇ ㄒㄧㄠˇ ㄉㄧㄥˋ ㄌㄧˇ

Фэйма сяодинли

/feɪ̯⁵¹ mä²¹⁴⁻³⁵ ɕi̯ɑʊ̯²¹⁴⁻²¹ tiŋ⁵¹ li²¹⁴⁻²¹⁽⁴⁾/

/fɐi̯³³ maː¹³ siːu̯³⁵ tɪŋ²² lei̯¹³/

/fɐi̯³³ maː¹³ siːu̯³⁵ tɪŋ²² lei̯¹³/

Fermat's little theorem
Fermat's little theorem


費馬大定理

ㄈㄟˋ ㄇㄚˇ ㄉㄚˋ ㄉㄧㄥˋ ㄌㄧˇ

Фэйма дадинли

/feɪ̯⁵¹ mä²¹⁴⁻²¹ tä⁵¹⁻⁵³ tiŋ⁵¹ li²¹⁴⁻²¹⁽⁴⁾/

/fɐi̯³³ maː¹³ taːi̯²² tɪŋ²² lei̯¹³/

/fɐi̯³³ maː¹³ taːi̯²² tɪŋ²² lei̯¹³/

Fermat's Last Theorem
Fermat's Last Theorem


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

en-noun

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)

Named after German mathematician Peter Gustav Lejeune Dirichlet.

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.


hữu tỉ

[hɨw˧˨ tɪj˧˨]

[hɨw˨˩˦ tɪj˨˩˦]

Sino-Vietnamese word from 有比, composed of 有 (“to have”) and 比 (“ratio”), perhaps a calque from a European cognate of English rational, ultimately from Latin ratiōnālis (“rational; reasonable”), from ratiō (“reason; ration; ratio”). Compare Japanese 有理 (yūri, literally “reasonable; rational”).


vô tỉ

Sino-Vietnamese word from 無比, composed of 無 (“no”) and 比 (“ratio”). See hữu tỉ. Compare Chinese 無比 /无比 (wúbǐ, “matchless; incomparable”) and Japanese 無理 (muri, “unreasonable; irrational”).


jäännösluokka

jäännösluokka (nominative singular)
jäännösluokat (nominative plural)
jäännösluokka (accusative nominative singular)
jäännösluokat (accusative nominative plural)
jäännösluokan (accusative genitive singular)
jäännösluokat (accusative genitive plural)
jäännösluokan (genitive singular)
jäännösluokkien (genitive plural)
jäännösluokkain (genitive plural rare)
jäännösluokkaa (partitive singular)
jäännösluokkia (partitive plural)
jäännösluokassa (inessive singular)
jäännösluokissa (inessive plural)
jäännösluokasta (elative singular)
jäännösluokista (elative plural)
jäännösluokkaan (illative singular)
jäännösluokkiin (illative plural)
jäännösluokalla (adessive singular)
jäännösluokilla (adessive plural)
jäännösluokalta (ablative singular)
jäännösluokilta (ablative plural)
jäännösluokalle (allative singular)
jäännösluokille (allative plural)
jäännösluokkana (essive singular)
jäännösluokkina (essive plural)
jäännösluokaksi (singular translative)
jäännösluokiksi (plural translative)
jäännösluokatta (abessive singular)
jäännösluokitta (abessive plural)
jäännösluokin (instructive plural)
See the possessive forms below. (comitative)
jäännösluokkani (nominative singular)
jäännösluokkani (nominative plural)
jäännösluokkani (accusative nominative singular)
jäännösluokkani (accusative nominative plural)
jäännösluokkani (accusative genitive singular)
jäännösluokkani (accusative genitive plural)
jäännösluokkani (genitive singular)
jäännösluokkieni (genitive plural)
jäännösluokkaini (genitive plural rare)
jäännösluokkaani (partitive singular)
jäännösluokkiani (partitive plural)
jäännösluokassani (inessive singular)
jäännösluokissani (inessive plural)
jäännösluokastani (elative singular)
jäännösluokistani (elative plural)
jäännösluokkaani (illative singular)
jäännösluokkiini (illative plural)
jäännösluokallani (adessive singular)
jäännösluokillani (adessive plural)
jäännösluokaltani (ablative singular)
jäännösluokiltani (ablative plural)
jäännösluokalleni (allative singular)
jäännösluokilleni (allative plural)
jäännösluokkanani (essive singular)
jäännösluokkinani (essive plural)
jäännösluokakseni (singular translative)
jäännösluokikseni (plural translative)
jäännösluokattani (abessive singular)
jäännösluokittani (abessive plural)
jäännösluokkineni (comitative plural)
jäännösluokkasi (nominative singular)
jäännösluokkasi (nominative plural)
jäännösluokkasi (accusative nominative singular)
jäännösluokkasi (accusative nominative plural)
jäännösluokkasi (accusative genitive singular)
jäännösluokkasi (accusative genitive plural)
jäännösluokkasi (genitive singular)
jäännösluokkiesi (genitive plural)
jäännösluokkaisi (genitive plural rare)
jäännösluokkaasi (partitive singular)
jäännösluokkiasi (partitive plural)
jäännösluokassasi (inessive singular)
jäännösluokissasi (inessive plural)
jäännösluokastasi (elative singular)
jäännösluokistasi (elative plural)
jäännösluokkaasi (illative singular)
jäännösluokkiisi (illative plural)
jäännösluokallasi (adessive singular)
jäännösluokillasi (adessive plural)
jäännösluokaltasi (ablative singular)
jäännösluokiltasi (ablative plural)
jäännösluokallesi (allative singular)
jäännösluokillesi (allative plural)
jäännösluokkanasi (essive singular)
jäännösluokkinasi (essive plural)
jäännösluokaksesi (singular translative)
jäännösluokiksesi (plural translative)
jäännösluokattasi (abessive singular)
jäännösluokittasi (abessive plural)
jäännösluokkinesi (comitative plural)
jäännösluokkamme (nominative singular)
jäännösluokkamme (nominative plural)
jäännösluokkamme (accusative nominative singular)
jäännösluokkamme (accusative nominative plural)
jäännösluokkamme (accusative genitive singular)
jäännösluokkamme (accusative genitive plural)
jäännösluokkamme (genitive singular)
jäännösluokkiemme (genitive plural)
jäännösluokkaimme (genitive plural rare)
jäännösluokkaamme (partitive singular)
jäännösluokkiamme (partitive plural)
jäännösluokassamme (inessive singular)
jäännösluokissamme (inessive plural)
jäännösluokastamme (elative singular)
jäännösluokistamme (elative plural)
jäännösluokkaamme (illative singular)
jäännösluokkiimme (illative plural)
jäännösluokallamme (adessive singular)
jäännösluokillamme (adessive plural)
jäännösluokaltamme (ablative singular)
jäännösluokiltamme (ablative plural)
jäännösluokallemme (allative singular)
jäännösluokillemme (allative plural)
jäännösluokkanamme (essive singular)
jäännösluokkinamme (essive plural)
jäännösluokaksemme (singular translative)
jäännösluokiksemme (plural translative)
jäännösluokattamme (abessive singular)
jäännösluokittamme (abessive plural)
jäännösluokkinemme (comitative plural)
jäännösluokkanne (nominative singular)
jäännösluokkanne (nominative plural)
jäännösluokkanne (accusative nominative singular)
jäännösluokkanne (accusative nominative plural)
jäännösluokkanne (accusative genitive singular)
jäännösluokkanne (accusative genitive plural)
jäännösluokkanne (genitive singular)
jäännösluokkienne (genitive plural)
jäännösluokkainne (genitive plural rare)
jäännösluokkaanne (partitive singular)
jäännösluokkianne (partitive plural)
jäännösluokassanne (inessive singular)
jäännösluokissanne (inessive plural)
jäännösluokastanne (elative singular)
jäännösluokistanne (elative plural)
jäännösluokkaanne (illative singular)
jäännösluokkiinne (illative plural)
jäännösluokallanne (adessive singular)
jäännösluokillanne (adessive plural)
jäännösluokaltanne (ablative singular)
jäännösluokiltanne (ablative plural)
jäännösluokallenne (allative singular)
jäännösluokillenne (allative plural)
jäännösluokkananne (essive singular)
jäännösluokkinanne (essive plural)
jäännösluokaksenne (singular translative)
jäännösluokiksenne (plural translative)
jäännösluokattanne (abessive singular)
jäännösluokittanne (abessive plural)
jäännösluokkinenne (comitative plural)
jäännösluokkansa (nominative singular)
jäännösluokkansa (nominative plural)
jäännösluokkansa (accusative nominative singular)
jäännösluokkansa (accusative nominative plural)
jäännösluokkansa (accusative genitive singular)
jäännösluokkansa (accusative genitive plural)
jäännösluokkansa (genitive singular)
jäännösluokkiensa (genitive plural)
jäännösluokkainsa (genitive plural rare)
jäännösluokkaansa (partitive singular)
jäännösluokkiaan (partitive plural)
jäännösluokkiansa (partitive plural)
jäännösluokassaan (inessive singular)
jäännösluokassansa (inessive singular)
jäännösluokissaan (inessive plural)
jäännösluokissansa (inessive plural)
jäännösluokastaan (elative singular)
jäännösluokastansa (elative singular)
jäännösluokistaan (elative plural)
jäännösluokistansa (elative plural)
jäännösluokkaansa (illative singular)
jäännösluokkiinsa (illative plural)
jäännösluokallaan (adessive singular)
jäännösluokallansa (adessive singular)
jäännösluokillaan (adessive plural)
jäännösluokillansa (adessive plural)
jäännösluokaltaan (ablative singular)
jäännösluokaltansa (ablative singular)
jäännösluokiltaan (ablative plural)
jäännösluokiltansa (ablative plural)
jäännösluokalleen (allative singular)
jäännösluokallensa (allative singular)
jäännösluokilleen (allative plural)
jäännösluokillensa (allative plural)
jäännösluokkanaan (essive singular)
jäännösluokkanansa (essive singular)
jäännösluokkinaan (essive plural)
jäännösluokkinansa (essive plural)
jäännösluokakseen (singular translative)
jäännösluokaksensa (singular translative)
jäännösluokikseen (plural translative)
jäännösluokiksensa (plural translative)
jäännösluokattaan (abessive singular)
jäännösluokattansa (abessive singular)
jäännösluokittaan (abessive plural)
jäännösluokittansa (abessive plural)
jäännösluokkineen (comitative plural)
jäännösluokkinensa (comitative plural)

/ˈjæːnːøsˌluo̯kːɑ/

[ˈjæːnːø̞s̠ˌluo̞̯kːɑ̝]


primitive element

en-noun

primitive element (plural primitive elements)

primitiivinen alkio

element of a coalgebra satisfying a particular condition

1972, W. Wesley Peterson, E. J. Weldon, Jr., Error-correcting Codes, The MIT Press, 2nd Edition, page 457,
Let A be a prime number for which 2 is a primitive element. Then 2ᴬ⁻¹-1 is divisible by A.

multiplicative field
multiplicative field

(number theory) Given a modulus n, a number g such that every number coprime to n is congruent (modulo n) to some power of g; equivalently, a generator of the multiplicative field of integers modulo n.


root of unity

In the case of the field of complex numbers, it follows from de Moivre's formula that the n nth roots of unity are #92;textstyle#92;cos#92;left(k#92;frac#123;2#92;pi#125;#123;n#125;#92;right)#43;i#92;sin#92;left(k#92;frac#123;2#92;pi#125;#123;n#125;#92;right), where k#61;1,#92;dots,n.

We now show that the primitive n-th roots of unity generate the other n-th roots of unity.

2001, Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, page 89:

A nice application of Hensel's Lemma is to determine which roots of unity can be found in #92;Q#95;p.

2003, Fernando Gouvêa, p-adic Numbers: An Introduction, Springer, page 72:

We have seen that, for a fixed value of n, the multiplicative group (U#95;n,#92;dot) is generated by any primitive nth root of unity. In particular, if #92;omega is a primitive 6th root of unity, then #92;omega⁶#61;1, six is the smallest positive integer for which this is true, and U#95;6#61;#92;#123;#92;omega⁰,#92;omega,#92;omega²,#92;omega³,#92;omega⁴,#92;omega⁵#92;#125;. It is easy to see that #92;omega², which is a 6th root of unity, is also a cube root of unity. The same is true of #92;omega⁴. The element #92;omega³ is a square root of unity, whereas #92;omega⁵ is primitive.

2007, Carl L. DeVito, Harmonic Analysis: A Gentle Introduction, Jones & Bartlett Learning, page 150:

(number theory) An element of a given field (especially, a complex number) x such that for some positive integer n, xⁿ = 1.

racine de l’unité

field element, some positive power of which equals 1


Smarandache function

Named after Florentin Smarandache, who rediscovered the function in 1980.

(number theory) A function, denoted by S(n) for some positive integer n, that yields the smallest number s such that n divides the factorial s!. For example, the number 8 does not divide 1!, 2!, 3!, but does divide 4!, so S(8) = 4.


Kronecker symbol

Kronecker symbol (plural Kronecker symbols)

(number theory) (a/n) or (a|n), a generalization of the Jacobi symbol to all integers n.


alkulukukaksoset

alkulukukaksoset (nominative plural)
alkulukukaksoset (accusative nominative plural)
alkulukukaksoset (accusative genitive plural)
alkulukukaksosten (genitive plural)
alkulukukaksosien (genitive plural)
alkulukukaksosia (partitive plural)
alkulukukaksosissa (inessive plural)
alkulukukaksosista (elative plural)
alkulukukaksosiin (illative plural)
alkulukukaksosilla (adessive plural)
alkulukukaksosilta (ablative plural)
alkulukukaksosille (allative plural)
alkulukukaksosina (essive plural)
alkulukukaksosiksi (plural translative)
alkulukukaksositta (abessive plural)
alkulukukaksosin (instructive plural)
See the possessive forms below. (comitative)
alkulukukaksoseni (nominative plural)
alkulukukaksoseni (accusative nominative plural)
alkulukukaksoseni (accusative genitive plural)
alkulukukaksosteni (genitive plural)
alkulukukaksosieni (genitive plural)
alkulukukaksosiani (partitive plural)
alkulukukaksosissani (inessive plural)
alkulukukaksosistani (elative plural)
alkulukukaksosiini (illative plural)
alkulukukaksosillani (adessive plural)
alkulukukaksosiltani (ablative plural)
alkulukukaksosilleni (allative plural)
alkulukukaksosinani (essive plural)
alkulukukaksosikseni (plural translative)
alkulukukaksosittani (abessive plural)
alkulukukaksosineni (comitative plural)
alkulukukaksosesi (nominative plural)
alkulukukaksosesi (accusative nominative plural)
alkulukukaksosesi (accusative genitive plural)
alkulukukaksostesi (genitive plural)
alkulukukaksosiesi (genitive plural)
alkulukukaksosiasi (partitive plural)
alkulukukaksosissasi (inessive plural)
alkulukukaksosistasi (elative plural)
alkulukukaksosiisi (illative plural)
alkulukukaksosillasi (adessive plural)
alkulukukaksosiltasi (ablative plural)
alkulukukaksosillesi (allative plural)
alkulukukaksosinasi (essive plural)
alkulukukaksosiksesi (plural translative)
alkulukukaksosittasi (abessive plural)
alkulukukaksosinesi (comitative plural)
alkulukukaksosemme (nominative plural)
alkulukukaksosemme (accusative nominative plural)
alkulukukaksosemme (accusative genitive plural)
alkulukukaksostemme (genitive plural)
alkulukukaksosiemme (genitive plural)
alkulukukaksosiamme (partitive plural)
alkulukukaksosissamme (inessive plural)
alkulukukaksosistamme (elative plural)
alkulukukaksosiimme (illative plural)
alkulukukaksosillamme (adessive plural)
alkulukukaksosiltamme (ablative plural)
alkulukukaksosillemme (allative plural)
alkulukukaksosinamme (essive plural)
alkulukukaksosiksemme (plural translative)
alkulukukaksosittamme (abessive plural)
alkulukukaksosinemme (comitative plural)
alkulukukaksosenne (nominative plural)
alkulukukaksosenne (accusative nominative plural)
alkulukukaksosenne (accusative genitive plural)
alkulukukaksostenne (genitive plural)
alkulukukaksosienne (genitive plural)
alkulukukaksosianne (partitive plural)
alkulukukaksosissanne (inessive plural)
alkulukukaksosistanne (elative plural)
alkulukukaksosiinne (illative plural)
alkulukukaksosillanne (adessive plural)
alkulukukaksosiltanne (ablative plural)
alkulukukaksosillenne (allative plural)
alkulukukaksosinanne (essive plural)
alkulukukaksosiksenne (plural translative)
alkulukukaksosittanne (abessive plural)
alkulukukaksosinenne (comitative plural)
alkulukukaksosensa (nominative plural)
alkulukukaksosensa (accusative nominative plural)
alkulukukaksosensa (accusative genitive plural)
alkulukukaksostensa (genitive plural)
alkulukukaksosiensa (genitive plural)
alkulukukaksosiaan (partitive plural)
alkulukukaksosiansa (partitive plural)
alkulukukaksosissaan (inessive plural)
alkulukukaksosissansa (inessive plural)
alkulukukaksosistaan (elative plural)
alkulukukaksosistansa (elative plural)
alkulukukaksosiinsa (illative plural)
alkulukukaksosillaan (adessive plural)
alkulukukaksosillansa (adessive plural)
alkulukukaksosiltaan (ablative plural)
alkulukukaksosiltansa (ablative plural)
alkulukukaksosilleen (allative plural)
alkulukukaksosillensa (allative plural)
alkulukukaksosinaan (essive plural)
alkulukukaksosinansa (essive plural)
alkulukukaksosikseen (plural translative)
alkulukukaksosiksensa (plural translative)
alkulukukaksosittaan (abessive plural)
alkulukukaksosittansa (abessive plural)
alkulukukaksosineen (comitative plural)
alkulukukaksosinensa (comitative plural)


alkutekijähajotelma

alkutekijähajotelma (nominative singular)
alkutekijähajotelmat (nominative plural)
alkutekijähajotelma (accusative nominative singular)
alkutekijähajotelmat (accusative nominative plural)
alkutekijähajotelman (accusative genitive singular)
alkutekijähajotelmat (accusative genitive plural)
alkutekijähajotelman (genitive singular)
alkutekijähajotelmien (genitive plural)
alkutekijähajotelmain (genitive plural rare)
alkutekijähajotelmaa (partitive singular)
alkutekijähajotelmia (partitive plural)
alkutekijähajotelmassa (inessive singular)
alkutekijähajotelmissa (inessive plural)
alkutekijähajotelmasta (elative singular)
alkutekijähajotelmista (elative plural)
alkutekijähajotelmaan (illative singular)
alkutekijähajotelmiin (illative plural)
alkutekijähajotelmalla (adessive singular)
alkutekijähajotelmilla (adessive plural)
alkutekijähajotelmalta (ablative singular)
alkutekijähajotelmilta (ablative plural)
alkutekijähajotelmalle (allative singular)
alkutekijähajotelmille (allative plural)
alkutekijähajotelmana (essive singular)
alkutekijähajotelmina (essive plural)
alkutekijähajotelmaksi (singular translative)
alkutekijähajotelmiksi (plural translative)
alkutekijähajotelmatta (abessive singular)
alkutekijähajotelmitta (abessive plural)
alkutekijähajotelmin (instructive plural)
See the possessive forms below. (comitative)
alkutekijähajotelmani (nominative singular)
alkutekijähajotelmani (nominative plural)
alkutekijähajotelmani (accusative nominative singular)
alkutekijähajotelmani (accusative nominative plural)
alkutekijähajotelmani (accusative genitive singular)
alkutekijähajotelmani (accusative genitive plural)
alkutekijähajotelmani (genitive singular)
alkutekijähajotelmieni (genitive plural)
alkutekijähajotelmaini (genitive plural rare)
alkutekijähajotelmaani (partitive singular)
alkutekijähajotelmiani (partitive plural)
alkutekijähajotelmassani (inessive singular)
alkutekijähajotelmissani (inessive plural)
alkutekijähajotelmastani (elative singular)
alkutekijähajotelmistani (elative plural)
alkutekijähajotelmaani (illative singular)
alkutekijähajotelmiini (illative plural)
alkutekijähajotelmallani (adessive singular)
alkutekijähajotelmillani (adessive plural)
alkutekijähajotelmaltani (ablative singular)
alkutekijähajotelmiltani (ablative plural)
alkutekijähajotelmalleni (allative singular)
alkutekijähajotelmilleni (allative plural)
alkutekijähajotelmanani (essive singular)
alkutekijähajotelminani (essive plural)
alkutekijähajotelmakseni (singular translative)
alkutekijähajotelmikseni (plural translative)
alkutekijähajotelmattani (abessive singular)
alkutekijähajotelmittani (abessive plural)
alkutekijähajotelmineni (comitative plural)
alkutekijähajotelmasi (nominative singular)
alkutekijähajotelmasi (nominative plural)
alkutekijähajotelmasi (accusative nominative singular)
alkutekijähajotelmasi (accusative nominative plural)
alkutekijähajotelmasi (accusative genitive singular)
alkutekijähajotelmasi (accusative genitive plural)
alkutekijähajotelmasi (genitive singular)
alkutekijähajotelmiesi (genitive plural)
alkutekijähajotelmaisi (genitive plural rare)
alkutekijähajotelmaasi (partitive singular)
alkutekijähajotelmiasi (partitive plural)
alkutekijähajotelmassasi (inessive singular)
alkutekijähajotelmissasi (inessive plural)
alkutekijähajotelmastasi (elative singular)
alkutekijähajotelmistasi (elative plural)
alkutekijähajotelmaasi (illative singular)
alkutekijähajotelmiisi (illative plural)
alkutekijähajotelmallasi (adessive singular)
alkutekijähajotelmillasi (adessive plural)
alkutekijähajotelmaltasi (ablative singular)
alkutekijähajotelmiltasi (ablative plural)
alkutekijähajotelmallesi (allative singular)
alkutekijähajotelmillesi (allative plural)
alkutekijähajotelmanasi (essive singular)
alkutekijähajotelminasi (essive plural)
alkutekijähajotelmaksesi (singular translative)
alkutekijähajotelmiksesi (plural translative)
alkutekijähajotelmattasi (abessive singular)
alkutekijähajotelmittasi (abessive plural)
alkutekijähajotelminesi (comitative plural)
alkutekijähajotelmamme (nominative singular)
alkutekijähajotelmamme (nominative plural)
alkutekijähajotelmamme (accusative nominative singular)
alkutekijähajotelmamme (accusative nominative plural)
alkutekijähajotelmamme (accusative genitive singular)
alkutekijähajotelmamme (accusative genitive plural)
alkutekijähajotelmamme (genitive singular)
alkutekijähajotelmiemme (genitive plural)
alkutekijähajotelmaimme (genitive plural rare)
alkutekijähajotelmaamme (partitive singular)
alkutekijähajotelmiamme (partitive plural)
alkutekijähajotelmassamme (inessive singular)
alkutekijähajotelmissamme (inessive plural)
alkutekijähajotelmastamme (elative singular)
alkutekijähajotelmistamme (elative plural)
alkutekijähajotelmaamme (illative singular)
alkutekijähajotelmiimme (illative plural)
alkutekijähajotelmallamme (adessive singular)
alkutekijähajotelmillamme (adessive plural)
alkutekijähajotelmaltamme (ablative singular)
alkutekijähajotelmiltamme (ablative plural)
alkutekijähajotelmallemme (allative singular)
alkutekijähajotelmillemme (allative plural)
alkutekijähajotelmanamme (essive singular)
alkutekijähajotelminamme (essive plural)
alkutekijähajotelmaksemme (singular translative)
alkutekijähajotelmiksemme (plural translative)
alkutekijähajotelmattamme (abessive singular)
alkutekijähajotelmittamme (abessive plural)
alkutekijähajotelminemme (comitative plural)
alkutekijähajotelmanne (nominative singular)
alkutekijähajotelmanne (nominative plural)
alkutekijähajotelmanne (accusative nominative singular)
alkutekijähajotelmanne (accusative nominative plural)
alkutekijähajotelmanne (accusative genitive singular)
alkutekijähajotelmanne (accusative genitive plural)
alkutekijähajotelmanne (genitive singular)
alkutekijähajotelmienne (genitive plural)
alkutekijähajotelmainne (genitive plural rare)
alkutekijähajotelmaanne (partitive singular)
alkutekijähajotelmianne (partitive plural)
alkutekijähajotelmassanne (inessive singular)
alkutekijähajotelmissanne (inessive plural)
alkutekijähajotelmastanne (elative singular)
alkutekijähajotelmistanne (elative plural)
alkutekijähajotelmaanne (illative singular)
alkutekijähajotelmiinne (illative plural)
alkutekijähajotelmallanne (adessive singular)
alkutekijähajotelmillanne (adessive plural)
alkutekijähajotelmaltanne (ablative singular)
alkutekijähajotelmiltanne (ablative plural)
alkutekijähajotelmallenne (allative singular)
alkutekijähajotelmillenne (allative plural)
alkutekijähajotelmananne (essive singular)
alkutekijähajotelminanne (essive plural)
alkutekijähajotelmaksenne (singular translative)
alkutekijähajotelmiksenne (plural translative)
alkutekijähajotelmattanne (abessive singular)
alkutekijähajotelmittanne (abessive plural)
alkutekijähajotelminenne (comitative plural)
alkutekijähajotelmansa (nominative singular)
alkutekijähajotelmansa (nominative plural)
alkutekijähajotelmansa (accusative nominative singular)
alkutekijähajotelmansa (accusative nominative plural)
alkutekijähajotelmansa (accusative genitive singular)
alkutekijähajotelmansa (accusative genitive plural)
alkutekijähajotelmansa (genitive singular)
alkutekijähajotelmiensa (genitive plural)
alkutekijähajotelmainsa (genitive plural rare)
alkutekijähajotelmaansa (partitive singular)
alkutekijähajotelmiaan (partitive plural)
alkutekijähajotelmiansa (partitive plural)
alkutekijähajotelmassaan (inessive singular)
alkutekijähajotelmassansa (inessive singular)
alkutekijähajotelmissaan (inessive plural)
alkutekijähajotelmissansa (inessive plural)
alkutekijähajotelmastaan (elative singular)
alkutekijähajotelmastansa (elative singular)
alkutekijähajotelmistaan (elative plural)
alkutekijähajotelmistansa (elative plural)
alkutekijähajotelmaansa (illative singular)
alkutekijähajotelmiinsa (illative plural)
alkutekijähajotelmallaan (adessive singular)
alkutekijähajotelmallansa (adessive singular)
alkutekijähajotelmillaan (adessive plural)
alkutekijähajotelmillansa (adessive plural)
alkutekijähajotelmaltaan (ablative singular)
alkutekijähajotelmaltansa (ablative singular)
alkutekijähajotelmiltaan (ablative plural)
alkutekijähajotelmiltansa (ablative plural)
alkutekijähajotelmalleen (allative singular)
alkutekijähajotelmallensa (allative singular)
alkutekijähajotelmilleen (allative plural)
alkutekijähajotelmillensa (allative plural)
alkutekijähajotelmanaan (essive singular)
alkutekijähajotelmanansa (essive singular)
alkutekijähajotelminaan (essive plural)
alkutekijähajotelminansa (essive plural)
alkutekijähajotelmakseen (singular translative)
alkutekijähajotelmaksensa (singular translative)
alkutekijähajotelmikseen (plural translative)
alkutekijähajotelmiksensa (plural translative)
alkutekijähajotelmattaan (abessive singular)
alkutekijähajotelmattansa (abessive singular)
alkutekijähajotelmittaan (abessive plural)
alkutekijähajotelmittansa (abessive plural)
alkutekijähajotelmineen (comitative plural)
alkutekijähajotelminensa (comitative plural)

[ˈɑ̝lkuˌt̪e̞k̟ijæˌhɑ̝jo̞ˌt̪e̞lmɑ̝]


neliövapaa

neliövapaa (nominative singular)
neliövapaat (nominative plural)
neliövapaa (accusative nominative singular)
neliövapaat (accusative nominative plural)
neliövapaan (accusative genitive singular)
neliövapaat (accusative genitive plural)
neliövapaan (genitive singular)
neliövapaiden (genitive plural)
neliövapaitten (genitive plural)
neliövapaata (partitive singular)
neliövapaita (partitive plural)
neliövapaassa (inessive singular)
neliövapaissa (inessive plural)
neliövapaasta (elative singular)
neliövapaista (elative plural)
neliövapaaseen (illative singular)
neliövapaisiin (illative plural)
neliövapaihin (illative plural rare)
neliövapaalla (adessive singular)
neliövapailla (adessive plural)
neliövapaalta (ablative singular)
neliövapailta (ablative plural)
neliövapaalle (allative singular)
neliövapaille (allative plural)
neliövapaana (essive singular)
neliövapaina (essive plural)
neliövapaaksi (singular translative)
neliövapaiksi (plural translative)
neliövapaatta (abessive singular)
neliövapaitta (abessive plural)
neliövapain (instructive plural)
neliövapaine (comitative plural)
neliövapaani (nominative singular)
neliövapaani (nominative plural)
neliövapaani (accusative nominative singular)
neliövapaani (accusative nominative plural)
neliövapaani (accusative genitive singular)
neliövapaani (accusative genitive plural)
neliövapaani (genitive singular)
neliövapaideni (genitive plural)
neliövapaitteni (genitive plural)
neliövapaatani (partitive singular)
neliövapaitani (partitive plural)
neliövapaassani (inessive singular)
neliövapaissani (inessive plural)
neliövapaastani (elative singular)
neliövapaistani (elative plural)
neliövapaaseeni (illative singular)
neliövapaisiini (illative plural)
neliövapaihini (illative plural rare)
neliövapaallani (adessive singular)
neliövapaillani (adessive plural)
neliövapaaltani (ablative singular)
neliövapailtani (ablative plural)
neliövapaalleni (allative singular)
neliövapailleni (allative plural)
neliövapaanani (essive singular)
neliövapainani (essive plural)
neliövapaakseni (singular translative)
neliövapaikseni (plural translative)
neliövapaattani (abessive singular)
neliövapaittani (abessive plural)
neliövapaineni (comitative plural)
neliövapaasi (nominative singular)
neliövapaasi (nominative plural)
neliövapaasi (accusative nominative singular)
neliövapaasi (accusative nominative plural)
neliövapaasi (accusative genitive singular)
neliövapaasi (accusative genitive plural)
neliövapaasi (genitive singular)
neliövapaidesi (genitive plural)
neliövapaittesi (genitive plural)
neliövapaatasi (partitive singular)
neliövapaitasi (partitive plural)
neliövapaassasi (inessive singular)
neliövapaissasi (inessive plural)
neliövapaastasi (elative singular)
neliövapaistasi (elative plural)
neliövapaaseesi (illative singular)
neliövapaisiisi (illative plural)
neliövapaihisi (illative plural rare)
neliövapaallasi (adessive singular)
neliövapaillasi (adessive plural)
neliövapaaltasi (ablative singular)
neliövapailtasi (ablative plural)
neliövapaallesi (allative singular)
neliövapaillesi (allative plural)
neliövapaanasi (essive singular)
neliövapainasi (essive plural)
neliövapaaksesi (singular translative)
neliövapaiksesi (plural translative)
neliövapaattasi (abessive singular)
neliövapaittasi (abessive plural)
neliövapainesi (comitative plural)
neliövapaamme (nominative singular)
neliövapaamme (nominative plural)
neliövapaamme (accusative nominative singular)
neliövapaamme (accusative nominative plural)
neliövapaamme (accusative genitive singular)
neliövapaamme (accusative genitive plural)
neliövapaamme (genitive singular)
neliövapaidemme (genitive plural)
neliövapaittemme (genitive plural)
neliövapaatamme (partitive singular)
neliövapaitamme (partitive plural)
neliövapaassamme (inessive singular)
neliövapaissamme (inessive plural)
neliövapaastamme (elative singular)
neliövapaistamme (elative plural)
neliövapaaseemme (illative singular)
neliövapaisiimme (illative plural)
neliövapaihimme (illative plural rare)
neliövapaallamme (adessive singular)
neliövapaillamme (adessive plural)
neliövapaaltamme (ablative singular)
neliövapailtamme (ablative plural)
neliövapaallemme (allative singular)
neliövapaillemme (allative plural)
neliövapaanamme (essive singular)
neliövapainamme (essive plural)
neliövapaaksemme (singular translative)
neliövapaiksemme (plural translative)
neliövapaattamme (abessive singular)
neliövapaittamme (abessive plural)
neliövapainemme (comitative plural)
neliövapaanne (nominative singular)
neliövapaanne (nominative plural)
neliövapaanne (accusative nominative singular)
neliövapaanne (accusative nominative plural)
neliövapaanne (accusative genitive singular)
neliövapaanne (accusative genitive plural)
neliövapaanne (genitive singular)
neliövapaidenne (genitive plural)
neliövapaittenne (genitive plural)
neliövapaatanne (partitive singular)
neliövapaitanne (partitive plural)
neliövapaassanne (inessive singular)
neliövapaissanne (inessive plural)
neliövapaastanne (elative singular)
neliövapaistanne (elative plural)
neliövapaaseenne (illative singular)
neliövapaisiinne (illative plural)
neliövapaihinne (illative plural rare)
neliövapaallanne (adessive singular)
neliövapaillanne (adessive plural)
neliövapaaltanne (ablative singular)
neliövapailtanne (ablative plural)
neliövapaallenne (allative singular)
neliövapaillenne (allative plural)
neliövapaananne (essive singular)
neliövapainanne (essive plural)
neliövapaaksenne (singular translative)
neliövapaiksenne (plural translative)
neliövapaattanne (abessive singular)
neliövapaittanne (abessive plural)
neliövapainenne (comitative plural)
neliövapaansa (nominative singular)
neliövapaansa (nominative plural)
neliövapaansa (accusative nominative singular)
neliövapaansa (accusative nominative plural)
neliövapaansa (accusative genitive singular)
neliövapaansa (accusative genitive plural)
neliövapaansa (genitive singular)
neliövapaidensa (genitive plural)
neliövapaittensa (genitive plural)
neliövapaataan (partitive singular)
neliövapaatansa (partitive singular)
neliövapaitaan (partitive plural)
neliövapaitansa (partitive plural)
neliövapaassaan (inessive singular)
neliövapaassansa (inessive singular)
neliövapaissaan (inessive plural)
neliövapaissansa (inessive plural)
neliövapaastaan (elative singular)
neliövapaastansa (elative singular)
neliövapaistaan (elative plural)
neliövapaistansa (elative plural)
neliövapaaseensa (illative singular)
neliövapaisiinsa (illative plural)
neliövapaihinsa (illative plural rare)
neliövapaallaan (adessive singular)
neliövapaallansa (adessive singular)
neliövapaillaan (adessive plural)
neliövapaillansa (adessive plural)
neliövapaaltaan (ablative singular)
neliövapaaltansa (ablative singular)
neliövapailtaan (ablative plural)
neliövapailtansa (ablative plural)
neliövapaalleen (allative singular)
neliövapaallensa (allative singular)
neliövapailleen (allative plural)
neliövapaillensa (allative plural)
neliövapaanaan (essive singular)
neliövapaanansa (essive singular)
neliövapainaan (essive plural)
neliövapainansa (essive plural)
neliövapaakseen (singular translative)
neliövapaaksensa (singular translative)
neliövapaikseen (plural translative)
neliövapaiksensa (plural translative)
neliövapaattaan (abessive singular)
neliövapaattansa (abessive singular)
neliövapaittaan (abessive plural)
neliövapaittansa (abessive plural)
neliövapaineen (comitative plural)
neliövapainensa (comitative plural)

/ˈneliøˌʋɑpɑː/

[ˈne̞liø̞ˌʋɑ̝pɑ̝ː]


resiprookkilaki

resiprookkilaki (nominative singular)
resiprookkilait (nominative plural)
resiprookkilaki (accusative nominative singular)
resiprookkilait (accusative nominative plural)
resiprookkilain (accusative genitive singular)
resiprookkilait (accusative genitive plural)
resiprookkilain (genitive singular)
resiprookkilakien (genitive plural)
resiprookkilakia (partitive singular)
resiprookkilakeja (partitive plural)
resiprookkilaissa (inessive singular)
resiprookkilaeissa (inessive plural)
resiprookkilaista (elative singular)
resiprookkilaeista (elative plural)
resiprookkilakiin (illative singular)
resiprookkilakeihin (illative plural)
resiprookkilailla (adessive singular)
resiprookkilaeilla (adessive plural)
resiprookkilailta (ablative singular)
resiprookkilaeilta (ablative plural)
resiprookkilaille (allative singular)
resiprookkilaeille (allative plural)
resiprookkilakina (essive singular)
resiprookkilakeina (essive plural)
resiprookkilaiksi (singular translative)
resiprookkilaeiksi (plural translative)
resiprookkilaitta (abessive singular)
resiprookkilaeitta (abessive plural)
resiprookkilaein (instructive plural)
See the possessive forms below. (comitative)
resiprookkilakini (nominative singular)
resiprookkilakini (nominative plural)
resiprookkilakini (accusative nominative singular)
resiprookkilakini (accusative nominative plural)
resiprookkilakini (accusative genitive singular)
resiprookkilakini (accusative genitive plural)
resiprookkilakini (genitive singular)
resiprookkilakieni (genitive plural)
resiprookkilakiani (partitive singular)
resiprookkilakejani (partitive plural)
resiprookkilaissani (inessive singular)
resiprookkilaeissani (inessive plural)
resiprookkilaistani (elative singular)
resiprookkilaeistani (elative plural)
resiprookkilakiini (illative singular)
resiprookkilakeihini (illative plural)
resiprookkilaillani (adessive singular)
resiprookkilaeillani (adessive plural)
resiprookkilailtani (ablative singular)
resiprookkilaeiltani (ablative plural)
resiprookkilailleni (allative singular)
resiprookkilaeilleni (allative plural)
resiprookkilakinani (essive singular)
resiprookkilakeinani (essive plural)
resiprookkilaikseni (singular translative)
resiprookkilaeikseni (plural translative)
resiprookkilaittani (abessive singular)
resiprookkilaeittani (abessive plural)
resiprookkilakeineni (comitative plural)
resiprookkilakisi (nominative singular)
resiprookkilakisi (nominative plural)
resiprookkilakisi (accusative nominative singular)
resiprookkilakisi (accusative nominative plural)
resiprookkilakisi (accusative genitive singular)
resiprookkilakisi (accusative genitive plural)
resiprookkilakisi (genitive singular)
resiprookkilakiesi (genitive plural)
resiprookkilakiasi (partitive singular)
resiprookkilakejasi (partitive plural)
resiprookkilaissasi (inessive singular)
resiprookkilaeissasi (inessive plural)
resiprookkilaistasi (elative singular)
resiprookkilaeistasi (elative plural)
resiprookkilakiisi (illative singular)
resiprookkilakeihisi (illative plural)
resiprookkilaillasi (adessive singular)
resiprookkilaeillasi (adessive plural)
resiprookkilailtasi (ablative singular)
resiprookkilaeiltasi (ablative plural)
resiprookkilaillesi (allative singular)
resiprookkilaeillesi (allative plural)
resiprookkilakinasi (essive singular)
resiprookkilakeinasi (essive plural)
resiprookkilaiksesi (singular translative)
resiprookkilaeiksesi (plural translative)
resiprookkilaittasi (abessive singular)
resiprookkilaeittasi (abessive plural)
resiprookkilakeinesi (comitative plural)
resiprookkilakimme (nominative singular)
resiprookkilakimme (nominative plural)
resiprookkilakimme (accusative nominative singular)
resiprookkilakimme (accusative nominative plural)
resiprookkilakimme (accusative genitive singular)
resiprookkilakimme (accusative genitive plural)
resiprookkilakimme (genitive singular)
resiprookkilakiemme (genitive plural)
resiprookkilakiamme (partitive singular)
resiprookkilakejamme (partitive plural)
resiprookkilaissamme (inessive singular)
resiprookkilaeissamme (inessive plural)
resiprookkilaistamme (elative singular)
resiprookkilaeistamme (elative plural)
resiprookkilakiimme (illative singular)
resiprookkilakeihimme (illative plural)
resiprookkilaillamme (adessive singular)
resiprookkilaeillamme (adessive plural)
resiprookkilailtamme (ablative singular)
resiprookkilaeiltamme (ablative plural)
resiprookkilaillemme (allative singular)
resiprookkilaeillemme (allative plural)
resiprookkilakinamme (essive singular)
resiprookkilakeinamme (essive plural)
resiprookkilaiksemme (singular translative)
resiprookkilaeiksemme (plural translative)
resiprookkilaittamme (abessive singular)
resiprookkilaeittamme (abessive plural)
resiprookkilakeinemme (comitative plural)
resiprookkilakinne (nominative singular)
resiprookkilakinne (nominative plural)
resiprookkilakinne (accusative nominative singular)
resiprookkilakinne (accusative nominative plural)
resiprookkilakinne (accusative genitive singular)
resiprookkilakinne (accusative genitive plural)
resiprookkilakinne (genitive singular)
resiprookkilakienne (genitive plural)
resiprookkilakianne (partitive singular)
resiprookkilakejanne (partitive plural)
resiprookkilaissanne (inessive singular)
resiprookkilaeissanne (inessive plural)
resiprookkilaistanne (elative singular)
resiprookkilaeistanne (elative plural)
resiprookkilakiinne (illative singular)
resiprookkilakeihinne (illative plural)
resiprookkilaillanne (adessive singular)
resiprookkilaeillanne (adessive plural)
resiprookkilailtanne (ablative singular)
resiprookkilaeiltanne (ablative plural)
resiprookkilaillenne (allative singular)
resiprookkilaeillenne (allative plural)
resiprookkilakinanne (essive singular)
resiprookkilakeinanne (essive plural)
resiprookkilaiksenne (singular translative)
resiprookkilaeiksenne (plural translative)
resiprookkilaittanne (abessive singular)
resiprookkilaeittanne (abessive plural)
resiprookkilakeinenne (comitative plural)
resiprookkilakinsa (nominative singular)
resiprookkilakinsa (nominative plural)
resiprookkilakinsa (accusative nominative singular)
resiprookkilakinsa (accusative nominative plural)
resiprookkilakinsa (accusative genitive singular)
resiprookkilakinsa (accusative genitive plural)
resiprookkilakinsa (genitive singular)
resiprookkilakiensa (genitive plural)
resiprookkilakiaan (partitive singular)
resiprookkilakiansa (partitive singular)
resiprookkilakejaan (partitive plural)
resiprookkilakejansa (partitive plural)
resiprookkilaissaan (inessive singular)
resiprookkilaissansa (inessive singular)
resiprookkilaeissaan (inessive plural)
resiprookkilaeissansa (inessive plural)
resiprookkilaistaan (elative singular)
resiprookkilaistansa (elative singular)
resiprookkilaeistaan (elative plural)
resiprookkilaeistansa (elative plural)
resiprookkilakiinsa (illative singular)
resiprookkilakeihinsa (illative plural)
resiprookkilaillaan (adessive singular)
resiprookkilaillansa (adessive singular)
resiprookkilaeillaan (adessive plural)
resiprookkilaeillansa (adessive plural)
resiprookkilailtaan (ablative singular)
resiprookkilailtansa (ablative singular)
resiprookkilaeiltaan (ablative plural)
resiprookkilaeiltansa (ablative plural)
resiprookkilailleen (allative singular)
resiprookkilaillensa (allative singular)
resiprookkilaeilleen (allative plural)
resiprookkilaeillensa (allative plural)
resiprookkilakinaan (essive singular)
resiprookkilakinansa (essive singular)
resiprookkilakeinaan (essive plural)
resiprookkilakeinansa (essive plural)
resiprookkilaikseen (singular translative)
resiprookkilaiksensa (singular translative)
resiprookkilaeikseen (plural translative)
resiprookkilaeiksensa (plural translative)
resiprookkilaittaan (abessive singular)
resiprookkilaittansa (abessive singular)
resiprookkilaeittaan (abessive plural)
resiprookkilaeittansa (abessive plural)
resiprookkilakeineen (comitative plural)
resiprookkilakeinensa (comitative plural)


จำนวนเฉพาะ

/t͡ɕam˧.nua̯n˧.t͡ɕʰa˨˩.pʰɔʔ˦˥/

From จำนวน (jam-nuuan) + เฉพาะ (chà-pɔ́).


သုဒ္ဒကိန်း

/θoʊʔda̰kéɪɴ/

သုဒ္ဒ (sudda.) + ကိန်း (kin:), the first element from Pali sudda, and the second element from Sanskrit केन्द्र (kendra).


密率

/mi⁵¹⁻⁵³ ly⁵¹/

/mɐt̚² lɵt̚²⁻³⁵/

/mi⁵¹⁻⁵³ ly⁵¹/

/mɐt̚² lɵt̚²⁻³⁵/


frumtala

is-noun

frumtala f (genitive singular frumtölu, nominative plural frumtölur)

frumtölu (genitive singular)
frumtölur (nominative plural)
frumtala (indefinite nominative singular)
frumtalan (definite nominative singular)
frumtölur (indefinite nominative plural)
frumtölurnar (definite nominative plural)
frumtölu (accusative indefinite singular)
frumtöluna (accusative definite singular)
frumtölur (accusative indefinite plural)
frumtölurnar (accusative definite plural)
frumtölu (dative indefinite singular)
frumtölunni (dative definite singular)
frumtölum (dative indefinite plural)
frumtölunum (dative definite plural)
frumtölu (genitive indefinite singular)
frumtölunnar (definite genitive singular)
frumtalna (genitive indefinite plural)
frumtalnanna (definite genitive plural)

From frum- (“primordial”) + tala (“number”).


sieve theory

(number theory) A set of general techniques for computing or estimating the size of sifted sets of integers.


almost prime

(number theory) A natural number for which there exists an absolute constant K such that the number has at most K prime factors.

Fastprimzahl

natural number with an absolute constant K


Bombieri-Friedlander-Iwaniec theorem

Proved in 1997 by John Friedlander and Henryk Iwaniec.

(number theory) A theorem stating that there are infinitely many prime numbers of the form a² + b⁴.


Chen's theorem

First stated by Chinese mathematician Chen Jingrun in 1966.

(number theory) A theorem stating that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime.


Selberg sieve

(number theory) A technique for estimating the size of sifted sets of positive integers that satisfy a set of conditions expressed by congruences.


nombor perdana

ms-noun

nombor perdana (plural nombor-nombor perdana, informal 1st possessive nombor perdanaku, 2nd possessive nombor perdanamu, 3rd possessive nombor perdananya)

nombor-nombor perdana (plural)
nombor perdana ku (first-person informal possessive)
nombor perdana mu (possessive second-person)
nombor perdana nya (possessive third-person)


輾轉相除法

ㄓㄢˇ ㄓㄨㄢˇ ㄒㄧㄤ ㄔㄨˊ ㄈㄚˇ

чжаньчжуаньсянчуфа

/ʈ͡ʂän²¹⁴⁻³⁵ ʈ͡ʂu̯än²¹⁴⁻²¹ ɕi̯ɑŋ⁵⁵ ʈ͡ʂʰu³⁵ fä²¹⁴⁻²¹⁽⁴⁾/

/t͡siːn³⁵ t͡syːn³⁵ sœːŋ⁵⁵ t͡sʰɵy̯²¹ faːt̚³/


Fermat number

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

named-after

(number theory) An integer which is one more than two raised to a power which is itself a power of two (i.e., is expressible in the form 2^(2ⁿ)+1 for some n>0); equivalently, a number that is one more than two raised to some power (is expressible as 2ⁿ+1) and is prime.


原根

/ɥɛn³⁵ kən⁵⁵/

/ɥɛn³⁵ kən⁵⁵/


Euclid's lemma

Named after ancient Greek mathematician Euclid of Alexandria (fl. 300 BCE). A version of the proposition appears in Book VII of his Elements.

I used Euclid's Lemma in a slightly sly way in the second chapter, where I ran through the argument that #92;sqrt 2 is irrational. I said there that if 2 is a factor of a² then a itself must be even. This follows from Euclid's Lemma upon taking p#61;2, the only even prime, and taking b#61;a. Indeed, using Euclid's Lemma it is not hard to generalize the argument showing #92;sqrt 2 to be irrational to prove that #92;sqrtp is irrational for any prime p.

1998, Peter M. Higgins, Mathematics for the Curious, Oxford University Press, page 78:

If a and b are not relatively prime, then the conclusion of Euclid's lemma may fail to hold. A specific example: 12#92;mid 9#92;cdot 8, but 12#92;nmid 9 and 12#92;nmid 8.

2007, David M. Burton, The History of Mathematics, McGraw-Hill, page 179:

In our discussion of Euclid's lemma (Corollary 2.18), we noted that the uniqueness of factorization of integers is a fact that we often take for granted given the way it is introduced in school.

2008, Martin Erickson, Anthony Vazzana, Introduction to Number Theory, Taylor & Francis (Chapman & Hall / CRC Press), page 42:

principal ideal domain
principal ideal domain

(number theory) The proposition that if a prime number p divides an arbitrary product ab of integers, then p divides a or b or both;

The proposition that if a prime number p divides an arbitrary product ab of integers, then p divides a or b or both;
slightly more generally, the proposition that for integers a, b, c, if a divides bc and gcd(a, b) = 1, then a divides c;
(algebra, by generalisation) the proposition that for elements a, b, c of a given principal ideal domain, if a divides bc and gcd(a, b) = 1, then a divides c.


Fundamentalsatz der Arithmetik

fundamental theorem of arithmetic
fundamental theorem of arithmetic

(number theory) fundamental theorem of arithmetic

Satz des Archimedes über Kugel und Kreiszylinder


fundamental theorem of arithmetic

en-noun

fundamental theorem of arithmetic (uncountable)

(number theory) the theorem that states that every integer greater than one is uniquely expressible as a product of prime numbers, which is called its prime factorization


aritmetikkens fundamentalsætning

Fundamentet bestod i aritmetikkens fundamentalsætning, der siger, at ethvert naturligt tal på entydig vis kan opløses i primfaktorer.

2008, Matematik for lærerstuderende - Omega, Samfundslitteratur, →ISBN, page 403:

The foundation was the fundamental theorem of arithmetic, which states that any natural number can be uniquely resolved into prime factors.

I alle ovenstående argumenter er Aritmetikkens Fundamentalsætning en nøgleingrediens.

2013, Hans Anton Salomonsen, Matematiske mysterier: Historien, forklaringerne og løsningerne, ISD LLC, →ISBN, page 196:

In all the above arguments, the fundamental theorem of arithmetic is a key ingredient.

Vi vil i dette afsnit se, hvorledes man ved hjælp af aritmetikkens fundamentalsætning for et vilkårligt naturligt tal n med kendt primfaktoropløsning dels kan bestemme divisorerne i n, dels antallet af divisorer i n, og dels summen af divisorerne i n.

2014, Gunnar Bomann, TAL OG ALGEBRA med historisk tilgang: Bog 2 Tal og algebra, BoD – Books on Demand, →ISBN, page 219:

In this section, we will see how, using the fundamental theorem of arithmetic, one can, for an arbitrary natural number n with a known prime-factor resolution can both determine the divisors of n, the number of divisors of n, and the sum of the divisors of n.


transcendental number theory

transcendental number theory (uncountable)

1975, Alan Baker, Transcendental Number Theory, Cambridge University Press (Cambridge Mathematical Library), page 48,
Nevertheless, transcendental number theory has led to new results in several associated subjects.

1998, Transcendental Number Theory and Related Topics: Proceedings of Japan-Korea Joint Seminar:

The transcendental number theory was originated by Liouville in 1844 when he showed a method to construct transcendental numbers.

2020, Saradha Natarajan, Ravindranathan Thangadurai, Pillars of Transcendental Number Theory, Springer, page 21:

transcendental number
transcendental number

(number theory) The branch of number theory that deals with transcendental numbers.


analytic number theory

1987, F. X. Gallagher, Applications of Guinand's Formula, A.C. Adolphson, A. Ghosh, J.B. Conrey, R.I. Yager (editors), Analytic Number Theory and Diophantine Problems: Proceedings of a Conference, Birkhäuser, page 135,
According to Weil [22], analytic number theory, which deals with inequalities and asymptotic formulas, is not number theory but analysis.

2002, Chaohua Jia, Kohji Matsumoto (editors), Analytic Number Theory, Kluwer Academic, page ix,
Talks covered various research fields including analytic number theory, algebraic number theory, modular forms and transcendental number theory.

Computational number theory, however, studies problems from elementary, algebraic geometric and analytic number theory which require the help of fast computers (particularly vector and parallel systems) and fast algorithms (particularly deterministic polynomial-time algorithms).

2002, Song Y. Yan, Number Theory for Computing, 2nd edition, Springer, page 174:

(mathematics, number theory) A branch of number theory that uses methods from mathematical analysis to solve problems about the integers.


abstract analytic number theory

abstract analytic number theory (uncountable)

The prime number theorem serves as a prototypical example of an idea generalised via abstract analytic number theory and adapted to other fields; the emphasis is on results pertaining to asymptotic analysis.

1975, John Knopfmacher (editor), Abstract Analytic Number Theory, North-Holland https://books.google.com.au/books?id=HIusKnF2kmEC&printsec=frontcover&dq=%22abstract+analytic+number+theory%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwjdm__u-5rsAhURIDQIHS14AjsQ6AEwAHoECAYQAg#v=onepage&q=%22abstract%20analytic%20number%20theory%22&f=false.

Only recently the authors completed the monograph [16] which contains a thorough presentation of the algebraic, combinatorial and analytic aspects of the theory of non-unique factorizations, together with self-contained introductions to additive group theory, to the theory of v-ideals and to abstract analytic number theory.

2006, Alfred Geroldinger, Franz Halter-Koch, “Non-unique factorizations: a survey”, in James W. Brewer, Sarah Glaz, William Heinzer, Bruce Olberding, editors, Multiplicative Ideal Theory in Commutative Algebra, Springer, page 207:

In [11], J. Knopfmacher gives an interesting exposition of abstract analytic number theory wherein many of the results of classical number theory are generalized in a suitable context of semigroups satisfying certain axioms.

2006, R. Sivaramakrishnan, Certain Number-Theoretic Episodes In Algebra, Taylor & Francis (Chapman & Hall/CRC), page 339:

analytic number theory
analytic number theory

(number theory) A branch of number theory in which suitable ideas and techniques from analytic number theory are generalised and applied to a variety of mathematical fields.