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# Fermat number

In mathematics a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form where n is a nonnegative integer. [1]

84 relations: Allan J. C. Cunningham, Amicable numbers, Édouard Lucas, Carl Friedrich Gauss, Christian Goldbach, Compass-and-straightedge construction, Composite number, Conjecture, Constructible polygon, Contradiction, Convergent series, Coprime integers, Corollary, Disquisitiones Arithmeticae, Distributed computing, Double exponential function, Euler's totient function, Expected value, Exponentiation by squaring, Fermat pseudoprime, Gaussian period, Gotthold Eisenstein, Heuristic argument, Infinity, Integer factorization, Irrational number, Ivan Pervushin, Jacobi symbol, James Cullen (mathematician), John Horton Conway, Lagrange's theorem (group theory), Landau's problems, Lenstra elliptic-curve factorization, Leonhard Euler, Linear congruential generator, List of sums of reciprocals, Lucas's theorem, Mathematical induction, Mathematical proof, Maurice Kraitchik, Megaprime, Mersenne prime, Modular arithmetic, Natural number, Necessity and sufficiency, On-Line Encyclopedia of Integer Sequences, Parity (mathematics), Pépin's test, Pierpont prime, Pierre de Fermat, ... Expand index (34 more) »

## Allan J. C. Cunningham

Allan Joseph Champneys Cunningham (1842 – 1928) was a British mathematician.

## Amicable numbers

Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number.

## Édouard Lucas

François Édouard Anatole Lucas (4 April 1842 – 3 October 1891) was a French mathematician.

## Carl Friedrich Gauss

Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.

## Christian Goldbach

Christian Goldbach (March 18, 1690 &ndash; November 20, 1764) was a German mathematician who also studied law.

## Compass-and-straightedge construction

Compass-and-straightedge construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass.

## Composite number

A composite number is a positive integer that can be formed by multiplying together two smaller positive integers.

## Conjecture

In mathematics, a conjecture is a conclusion or proposition based on incomplete information, for which no proof has been found.

## Constructible polygon

In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge.

In classical logic, a contradiction consists of a logical incompatibility between two or more propositions.

## Convergent series

In mathematics, a series is the sum of the terms of an infinite sequence of numbers.

## Coprime integers

In number theory, two integers and are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1.

## Corollary

A corollary is a statement that follows readily from a previous statement.

## Disquisitiones Arithmeticae

The Disquisitiones Arithmeticae (Latin for "Arithmetical Investigations") is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24.

## Distributed computing

Distributed computing is a field of computer science that studies distributed systems.

## Double exponential function

A double exponential function is a constant raised to the power of an exponential function.

## Euler's totient function

In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to.

## Expected value

In probability theory, the expected value of a random variable, intuitively, is the long-run average value of repetitions of the experiment it represents.

## Exponentiation by squaring

In mathematics and computer programming, exponentiating by squaring is a general method for fast computation of large positive integer powers of a number, or more generally of an element of a semigroup, like a polynomial or a square matrix.

## Fermat pseudoprime

In number theory, the Fermat pseudoprimes make up the most important class of pseudoprimes that come from Fermat's little theorem.

## Gaussian period

In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity.

## Gotthold Eisenstein

Ferdinand Gotthold Max Eisenstein (16 April 1823 – 11 October 1852) was a German mathematician.

## Heuristic argument

A heuristic argument is an argument that reasons from the value of a method or principle that has been shown by experimental (especially trial-and-error) investigation to be a useful aid in learning, discovery and problem-solving.

## Infinity

Infinity (symbol) is a concept describing something without any bound or larger than any natural number.

## Integer factorization

In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers.

## Irrational number

In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers.

## Ivan Pervushin

Ivan Mikheevich Pervushin (Иван Михеевич Первушин, sometimes transliterated as Pervusin or Pervouchine) (—) was a Russian clergyman and mathematician of the second half of the 19th century, known for his achievements in number theory.

## Jacobi symbol

Jacobi symbol for various k (along top) and n (along left side).

## James Cullen (mathematician)

Father James Cullen, S.J. (19 April 1867 &ndash; 7 December 1933) was born at Drogheda, County Louth, Ireland.

## John Horton Conway

John Horton Conway FRS (born 26 December 1937) is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory.

## Lagrange's theorem (group theory)

Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order (number of elements) of every subgroup H of G divides the order of G. The theorem is named after Joseph-Louis Lagrange.

## Landau's problems

At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about primes.

## Lenstra elliptic-curve factorization

The Lenstra elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization, which employs elliptic curves.

## Leonhard Euler

Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.

## Linear congruential generator

A linear congruential generator (LCG) is an algorithm that yields a sequence of pseudo-randomized numbers calculated with a discontinuous piecewise linear equation.

## List of sums of reciprocals

In mathematics and especially number theory, the sum of reciprocals generally is computed for the reciprocals of some or all of the positive integers (counting numbers)&mdash;that is, it is generally the sum of unit fractions.

## Lucas's theorem

In number theory, Lucas's theorem expresses the remainder of division of the binomial coefficient \tbinom by a prime number p in terms of the base p expansions of the integers m and n. Lucas's theorem first appeared in 1878 in papers by Édouard Lucas.

## Mathematical induction

Mathematical induction is a mathematical proof technique.

## Mathematical proof

In mathematics, a proof is an inferential argument for a mathematical statement.

## Maurice Kraitchik

Maurice Kraitchik (April 21, 1882 – August 19, 1957) was a Belgian mathematician and populariser.

## Megaprime

A megaprime is a prime number with at least one million decimal digits (whereas titanic prime is a prime number with at least 1,000 digits, and gigantic prime has at least 10,000 digits).

## Mersenne prime

In mathematics, a Mersenne prime is a prime number that is one less than a power of two.

## Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli).

## Natural number

In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").

## Necessity and sufficiency

In logic, necessity and sufficiency are terms used to describe an implicational relationship between statements.

## On-Line Encyclopedia of Integer Sequences

The On-Line Encyclopedia of Integer Sequences (OEIS), also cited simply as Sloane's, is an online database of integer sequences.

## Parity (mathematics)

In mathematics, parity is the property of an integer's inclusion in one of two categories: even or odd.

## Pépin's test

In mathematics, Pépin's test is a primality test, which can be used to determine whether a Fermat number is prime.

## Pierpont prime

A Pierpont prime is a prime number of the form for some nonnegative integers and.

## Pierre de Fermat

Pierre de Fermat (Between 31 October and 6 December 1607 – 12 January 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality.

## Pierre Wantzel

Pierre Laurent Wantzel (5 June 1814 in Paris – 21 May 1848 in Paris) was a French mathematician who proved that several ancient geometric problems were impossible to solve using only compass and straightedge.

## Primality test

A primality test is an algorithm for determining whether an input number is prime.

## Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

## Prime number theorem

In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers.

## Prime Pages

The Prime Pages is a website about prime numbers maintained by Chris Caldwell at the University of Tennessee at Martin.

## PrimeGrid

PrimeGrid is a volunteer distributed computing project searching for prime numbers of world-record size.

## Primitive root modulo n

In modular arithmetic, a branch of number theory, a number g is a primitive root modulo n if every number a coprime to n is congruent to a power of g modulo n. That is, for every integer a coprime to n, there is an integer k such that gk ≡ a (mod n).

## Probability

Probability is the measure of the likelihood that an event will occur.

## Probable prime

In number theory, a probable prime (PRP) is an integer that satisfies a specific condition that is satisfied by all prime numbers, but which is not satisfied by most composite numbers.

## Proth number

In number theory, a Proth number, named after the mathematician François Proth, is a number of the form where k is an odd positive integer and n is a positive integer such that 2^n > k. Without the latter condition, all odd integers greater than 1 would be Proth numbers.

## Proth's theorem

In number theory, Proth's theorem is a primality test for Proth numbers.

## Pseudoprime

A pseudoprime is a probable prime (an integer that shares a property common to all prime numbers) that is not actually prime.

## Pseudorandomness

A pseudorandom process is a process that appears to be random but is not.

## Pythagorean prime

A Pythagorean prime is a prime number of the form 4n + 1.

In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that: Otherwise, q is called a quadratic nonresidue modulo n. Originally an abstract mathematical concept from the branch of number theory known as modular arithmetic, quadratic residues are now used in applications ranging from acoustical engineering to cryptography and the factoring of large numbers.

## Randomness

Randomness is the lack of pattern or predictability in events.

## RANDU

RANDU is a linear congruential pseudorandom number generator of the Park–Miller type, which has been used since the 1960s.

## Recurrence relation

In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given: each further term of the sequence or array is defined as a function of the preceding terms.

## Sierpinski number

In number theory, a Sierpinski or Sierpiński number is an odd natural number k such that k \times 2^n + 1 is composite, for all natural numbers n. In 1960, Wacław Sierpiński proved that there are infinitely many odd integers k which have this property.

## Sign (mathematics)

In mathematics, the concept of sign originates from the property of every non-zero real number of being positive or negative.

## Solomon W. Golomb

Solomon Wolf Golomb (May 30, 1932 – May 1, 2016) was an American mathematician, engineer, and professor of electrical engineering at the University of Southern California, best known for his works on mathematical games.

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

## Square-free integer

In mathematics, a square-free integer is an integer which is divisible by no perfect square other than 1.

## Strong pseudoprime

In number theory, a probable prime is a number that passes a primality test.

## Sylvester's sequence

In number theory, Sylvester's sequence is an integer sequence in which each member of the sequence is the product of the previous members, plus one.

## Thomas Clausen (mathematician)

Thomas Clausen (January 16, 1801, Snogbæk, Sottrup Municipality, Duchy of Schleswig (now Denmark) &ndash; May 23, 1885, Derpt, Imperial Russia (now Estonia)) was a Danish mathematician and astronomer.

## Time complexity

In computer science, the time complexity is the computational complexity that describes the amount of time it takes to run an algorithm.

## Universal quantification

In predicate logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all".

## Wieferich prime

In number theory, a Wieferich prime is a prime number p such that p2 divides, therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides.

## 17 (number)

17 (seventeen) is the natural number following 16 and preceding 18.

## 257 (number)

257 (two hundred fifty-seven) is the natural number following 256 and preceding 258.

## 3

3 (three) is a number, numeral, and glyph.

## 5

5 (five) is a number, numeral, and glyph.

## 65,537

65537 is the integer after 65536 and before 65538.

## References

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