primitive element

en-noun

primitive element (plural primitive elements)

primitiivinen alkio

element of a coalgebra satisfying a particular condition

Let A be a prime number for which 2 is a primitive element. Then 2#123;A-1#125;-1 is divisible by A.

1972, W. Wesley Peterson, E. J. Weldon, Jr., Error-correcting Codes, 2nd edition, The MIT Press, page 457:

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.


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.


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.


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.


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


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:

mathematical analysis
mathematical analysis

(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.


additive number theory

Abstractly, additive number theory includes the study of abelian groups and commutative semigroups with an addition operation.

Two principal objects of study in additive number theory are the sumset A#43;B#61;#92;#123;a#43;b#58;a#92;inA,b#92;inB#92;#125; of two subsets A and B of elements from an abelian group G, and the h-fold sumset of A, hA#61;#92;underset#123;h#125;#123;#92;underbrace#123;A#43;#92;cdots#43;A

Additive number theory has close ties to combinatorial number theory and the geometry of numbers.

1966 [Macmillan], W. E. Deskins, Abstract Algebra, 1995, Dover, page 152,
One of the famous theorems of additive number theory states that each positive integer is expressible in at least one way as the sum of the squares of not more than four positive integers.

The Circle Method is a beautiful idea for studying many problems in additive number theory. It originated in investigations by Hardy and Ramanujan [HR] on the partition function P(n), the number of ways we can write n as a sum of positive integers. Since then it has been used to study problems in additive number theory ranging from writing numbers as sums of primes or k#92;mathsf#123;th#125; powers (for fixed k) to trying to count how many twin primes there are less than x.

2006, Steven J. Miller, Ramin Takloo-Bighash, An Invitation to Modern Number Theory, Princeton University Press, page 303:

Additive number theory is a very broad area within which lies the theory of partitions, and the subject of representation of integers as sums of squares, cubes, etc. There is a vast literature in additive number theory on problems of the following type: Given a subset A of the integers, when does it form a basis for the integers? That is, when can every positive integer be expressed as the sum of at most a (fixed) bounded number of elements of A?

2008, Krishnaswami Alladi, editor, Surveys in Number Theory, Springer, page ix:

(number theory) The subfield of number theory concerning the behaviour of sumsets (integer sets added to each other elementwise).


geometry of numbers

The field was initiated by German mathematician Hermann Minkowski (1910, Geometrie der Zahlen).

A typical approach in the geometry of numbers is to view a ring of algebraic integers as a lattice in #92;textstyle#92;mathbb#123;R#125;ⁿ; the study of these lattices provides fundamental information on algebraic numbers.

1969, C. G. Lekkerkerker, Geometry of Numbers, Wolters-Noordoff, North-Holland, page 1,
The geometry of numbers to which this book is devoted deals with arbitrary bodies and arbitrary lattices in the n-dimensional euclidean space. Its aim is to study various quantities describing the behaviour of a body with respect to a lattice.

2000, C. D. Olds, Anneli Lax, Giuliana Davidoff, The Geometry of Numbers, Mathematical Association of America:

2006, Enrico Bombieri, Walter Gubler, Heights in Diophantine Geometry, Cambridge University Press, page 181,
The easy proof is obtained applying the pigeon-hole principle to
α₁x_1+…+αₙx_n(mod 1)|x_i=0,…N,
or by geometry of numbers by applying Minkowski's first theorem in C.2.19 to the symmetric convex body of volume 2ⁿ⁺¹ given by
|X_0+α₁X_1+…αₙX_n|<N⁻ⁿ,|X_i|<N,i=1,…N.

(number theory) The subbranch of number theory which applies techniques from geometry to the study of algebraic numbers.


Chinese remainder theorem

the Chinese remainder theorem (canonical)

The earliest known formulation of the theorem was in the Chinese Sun Zi's Mathematical Manual circa 400 AD.

(number theory) A theorem stating that, if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime.

Chinese remainder theorem
Sunzi Suanjing


computational number theory

Topics studied in computational number theory include algorithms for primality testing and integer factorisation, finding solutions to Diophantine equations and explicit methods in arithmetic geometry.

Computational number theory has applications in cryptography, including RSA, elliptic curve cryptography and post-quantum cryptography, and is used to investigate conjectures and open problems in number theory, including the Riemann hypothesis.

1990, Carl Pomerance, editor, Proceedings of Symposia in Applied Mathematics, Volume 42: Cryptology and Computational Number Theory, American Mathematical Society:

1994, Hari Krishna, Bal Krishna, Kuo-Yu Lin, Jenn-Dong Sun, Computational Number Theory and Digital Signal Processing: Fast Algorithms and Error Control Techniques, CRC Press, https://books.google.com.au/books?id=8u1kx3peO6AC&printsec=frontcover&dq=%22computational+number+theory%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwiy0c_SnvjuAhUCyjgGHUIkBnUQ6AEwB3oECCUQAg#v=onepage&q=%22computational%20number%20theory%22&f=false.

2001, Kazimierz Alster, Jerzy Urbanowicz, Hugh C. Williams, editors, Public-Key Cryptography and Computational Number Theory: Proceedings of the International Conference, Walter de Gruyter:

(mathematics, number theory) The field of study dealing with computational methods for investigating and solving problems in number theory and arithmetic geometry.

théorie algorithmique des nombres

field of study dealing with computational methods in number theory


antiprime

highly composite number
highly composite number#English

(number theory) Synonym of highly composite number.


p-adic order

(number theory) Of a positive integer n, the exponent of the highest power of p that divides n. Often denoted νₚ(n). For example, ν₂(48)=4.


lifting the exponent

(number theory) A lemma used to calculate the p-adic order of differences and sums of perfect powers.


Polignac's conjecture

(number theory) The conjecture that, for any positive even number n, there are infinitely many prime gaps of size n.


Romanov's theorem

(number theory) A theorem stating that, given a fixed base b, the set of numbers that are the sum of a prime and a positive integer power of b has a positive lower asymptotic density.


Dickson's conjecture

Introduced by Leonard Eugene Dickson in 1904.

(number theory) The conjecture that, for a finite set of linear forms a₁ + b₁n, a₂ + b₂n, ..., aₖ + bₖn with bᵢ ≥ 1, there are infinitely many positive integers n for which they are all prime, unless there is a congruence condition preventing this.


Elliott-Halberstam conjecture

the Elliott-Halberstam conjecture (canonical)

Named for Peter D. T. A. Elliott and Heini Halberstam, who stated the conjecture in 1968.

arithmetic progression
arithmetic progression

(number theory) A conjecture about the distribution of prime numbers in arithmetic progressions, having many applications in sieve theory.


Schinzel's hypothesis H

(number theory) A famous open problem in mathematics, the hypothesis stating that, for every finite collection f_1,f_2,…,f_k of non-constant irreducible polynomials over the integers with positive leading coefficients, one of the following conditions holds: (i) there are infinitely many positive integers n such that all of f_1(n),f_2(n),…,f_k(n) are simultaneously prime numbers, or (ii) there is an integer m>1 (called a fixed divisor) which always divides the product f_1(n)f_2(n)⋯f_k(n).


Szpiro's conjecture

Named for Lucien Szpiro, who formulated it in the 1980s.

(number theory) A conjecture relating to the conductor and the discriminant of an elliptic curve, equivalent in a slightly modified form to the abc conjecture.


Brocard's conjecture

(number theory) The conjecture that there are at least four prime numbers between (pₙ)² and (pₙ₊₁)², where pₙ is the nth prime number, for every n ≥ 2.


Wall-Sun-Sun prime

Wall-Sun-Sun prime (plural Wall-Sun-Sun primes)

Wall-Sun-Sun primes (plural)
Wall-Sun-Sun prime number (alternative)

Named after American mathematician Donald Dines Wall and Chinese mathematicians Sun Zhihong and Sun Zhiwei, who have all contributed to the study of such primes.

Wall-Sun-Sun primes are conjectured to exist, but no example has yet been found.

Not a single Wall-Sun-Sun prime 5lt;plt;1.6#92;times 10#123;12#125; exists [McIntosh 1995]. Carry out a search for Wall-Sun-Sun primes somewhere above this limit.

1996, Richard E. Crandall, Topics in Advanced Scientific Computation, Springer, page 113:

Crandall et al. called in [56] such a prime number p satisfying p²#92;vertF#95;#123;p-(#92;fracp 5)#125; a Wall–Sun–Sun prime. There is no known example of a Wall–Sun–Sun prime and the congruence F#95;#123;p-(#92;fracp 5)#125;#92;equiv 0#92;pmod#123;p²#125;, can only be checked through explicit powering computations.

2020, Dorin Andrica, Ovidiu Bagdasar, Recurrent Sequences, Springer, page 88:

(number theory) A (hypothetical) prime number p such that p² divides F_π(p), where F_n is the Fibonacci sequence and π(p) is the pth Pisano period (the period length of the Fibonacci sequence reduced modulo p).

Donald Dines Wall
Sun Zhihong
Sun Zhiwei
Wall–Sun–Sun prime

nombre premier de Wall-Sun-Sun

conjectured prime number of a certain type


rank of apparition

rank of apparition (plural ranks of apparition)

Let (W#95;n) be an EDS^([Elliptic Divisibility Sequence]) associated to an elliptic curve E#47;K and a nonzero point P#92;inE(K) of finite order. Let r#92;ge 2 be the smallest index such that W#95;r#61;0. (The number r is called the rank of apparition of the sequence.)
[…]
Suppose that K is a finite field and that the rank of apparition r of (W#95;n) is at least 4.

1986, Joseph H. Silverman, The Arithmetic of Elliptic Curves, Springer, page 114:

1996, D. L. Wells, Residue Counts Modulo Three for the Fibonacci Triangle, G. E. Bergum, Andreas N. Philippou, Alwyn F. Horadam (editors), Applications of Fibonacci Numbers, Kluwer Academic, Softcover reprint, page 535,
A similar identification between Pascal's Triangle modulo p and the Fibonacci Triangle modulo p can be made for primes p which have the length of the period equal to twice their rank of apparition in the Fibonacci Sequence.

Indeed, as shown in [18, Theorem 4.27], there exist sequences #92;#123;U#95;n#92;#125; and primes p for which p has three ranks of apparition. In the previous section, we showed that if p#61;2, then p has no more than two ranks of apparition in #92;#123;U#95;n#92;#125;.

2013, E. L. Roettger, H. C. Williams, R. K. Guy, “Some Extensions of the Lucas Functions”, in Jonathan M. Borwein, Igor Shparlinski, Wadim Zudilin, editors, Number Theory and Related Fields, Springer, page 303:

divisibility sequence
divisibility sequence

(number theory) Given a positive integer m and a divisibility sequence Sₖ, the smallest index k such that Sₖ is divisible by m;

Given a positive integer m and a divisibility sequence Sₖ, the smallest index k such that Sₖ is divisible by m;
such an index for some generalisation of the concept (for example to allow multiple ranks of apparition for a given m).


Cramér's conjecture

Formulated by the Swedish mathematician Harald Cramér in 1936.

(number theory) An estimate for the size of gaps between consecutive prime numbers, stating that p_n+1-p_n=O(( log p_n)²), where pₙ denotes the nth prime number, O is big O notation, and log is the natural logarithm.


Green-Tao theorem

Proved by Ben Green and Terence Tao in 2004.

(number theory) A theorem stating that the sequence of prime numbers contains arbitrarily long arithmetic progressions.


Pólya's conjecture

Posited in 1919 by the Hungarian mathematician George Pólya.

(number theory) The conjecture (disproved in 1958) that at least half of the natural numbers smaller than any given number have an odd number of prime factors.


Hardy-Littlewood circle method

the Hardy-Littlewood circle method (canonical)

Named after G. H. Hardy and J. E. Littlewood, who developed the technique.

(number theory) A method used to compute the residues about zero for the generating function of a series, as part of proving asymptotic behaviour of the series.


Waring's problem

The problem was proposed in 1770 by Edward Waring.

(number theory) The problem of whether each natural number k has an associated positive integer s such that every natural number is the sum of at most s natural numbers raised to the power k.


magic constant

magic constant (countable and uncountable, plural magic constants)

(mathematics, number theory) The sum of numbers in any row, column, or diagonal of a magic square.


pandiagonal magic square

pandiagonal magic square (plural pandiagonal magic squares)

recreational mathematics
recreational mathematics

(number theory, recreational mathematics) A magic square with the additional property that the broken diagonals also summate to the magic constant.


broken diagonal

broken diagonal (plural broken diagonals)

(mathematics, number theory) In the theory of magic squares, a set of n cells forming two parallel diagonal lines in the square.


most-perfect magic square

most-perfect magic square (plural most-perfect magic squares)

(number theory, recreational mathematics) A magic square of order n containing the numbers 1 to n² and where each 2 × 2 subsquare sums to 2s, where s = n² + 1, and all pairs of integers distant n/2 along a (major) diagonal sum to s.


prime reciprocal magic square

prime reciprocal magic square (plural prime reciprocal magic squares)

(number theory, recreational mathematics) A magic square using the decimal digits of the reciprocal of a prime number.


multimagic square

multimagic square (plural multimagic squares)

(number theory, recreational mathematics) A magic square that remains magic even if all its numbers are replaced by their kth powers for 1 ≤ k ≤ P.


bimagic square

(number theory, mathematics) A magic square that remains magic when all of its numbers are replaced by their squares.


trimagic square

trimagic square (plural trimagic squares)

(number theory, recreational mathematics) A magic square that remains magic when all of its numbers are replaced by their cubes.


Pell number

(mathematics, number theory) An infinite sequence of integers that comprises the denominators of the closest rational approximations to the square root of 2.


Dirichlet character

Dirichlet character (plural Dirichlet characters)

1) χ(ab)=χ(a)χ(b);

2) χ(a)=0if; gcd (a,m)>1\ ne 0if; gcd (a,m)=1. (gcd is the greatest common divisor)

3) χ(a+m)=χ(a).

(analytic number theory) A complex-valued arithmetic function χ:ℤ→ℂ that satisfies the following conditions for some positive integer m and all integers a and b:


Gaussian moat

If we can keep finding 'fatter and fatter' Gaussian moats surrounding the origin, then the prime-hiker will have to keep taking longer and longer steps.

2006, Deanna Haunsperger, Stephen Kennedy, The Edge of the Universe: Celebrating Ten Years of Math Horizons, page 49:

The approach we follow is composed by three main components, each one defined by its own algorithm. The first one generates the necessary Gaussian primes, the second one calculates the Gaussian Moat, and the third one calculates the shortest paths.

2020, Hector Florez, Sanjay Misra, Applied Informatics: Third International Conference, ICAI 2020, Ota, Nigeria, October 29–31, 2020, Proceedings, page 483:

The Gaussian moat problem was posed originally by Basil Gordon in 1962 at the International Congress of Mathematicians held in Stockholm, […]

2022, Luca Dal Negro, Waves in Complex Media, page 372:

(number theory) The maximum difference between consecutive numbers in a sequence of distinct Gaussian prime numbers.


Stern-Brocot tree

Stern-Brocot tree (plural Stern-Brocot trees)

Introduced independently by Moritz Stern (1858) and Achille Brocot (1861).

(number theory) An infinite complete binary tree whose vertices correspond one-for-one to the positive rational numbers, whose values are ordered from the left to the right as in a search tree.


narcissistic number

narcissistic number (plural narcissistic numbers)

(number theory) A number that, in a given base, is the sum of its own digits each raised to the power of the number of digits.


Armstrong number

Armstrong number (plural Armstrong numbers)


Ramanujan's sum

Ramanujan's sum (plural Ramanujan's sums)

Vinogradov's theorem
Vinogradov's theorem

(number theory) A function, usually denoted c_q(n), of two positive integer variables q and n, defined by the formula c_q(n)=∑_(1<a≤q atop (a,q)=1)e^(2πi tfrac aqn),, where (a, q) = 1 means that a only takes on values coprime to q. It is used in the proof of Vinogradov's theorem.


Vinogradov's theorem

Named after Ivan Matveyevich Vinogradov, who proved it in the 1930s.

Goldbach's weak conjecture
Goldbach's weak conjecture

(number theory) A result which implies that any sufficiently large odd integer can be written as a sum of three prime numbers; a weaker form of Goldbach's weak conjecture.


Legendre's conjecture

(number theory) A conjecture stating that there is a prime number between n² and (n+1)² for every positive integer n.


twin prime conjecture

(number theory) A conjecture stating that there are infinitely many twin primes.


Pisano period

Named after Leonardo Bigollo Pisano a.k.a. Fibonacci (c. 1170 – c. 1240–1250).

(number theory) The period (of repetition) of the sequence of Fibonacci numbers reduced modulo a given natural number (≥1); it can be considered to be a function π(n) where n is the modulus of reduction (see modular arithmetic) and the function returns the period (of the reduced Fibonacci sequence) corresponding to that given modulus.


snowball prime

(mathematics, number theory, uncommon) A prime number which remains a prime at every stage as it is written and truncated from the left.


rough number

(number theory) A k-rough number; a positive integer whose prime factors are all greater than or equal to k.