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Positive integer of the form (2^(2^n))+1
In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a positive integer of the form: F
Fermat_number
French mathematician and lawyer (1601–1665)
Pierre de Fermat (/fɜːrˈmɑː/; French: [pjɛʁ də fɛʁma]; 17 August 1601 – 12 January 1665) was a French magistrate, polymath, and above all, a mathematician
Pierre_de_Fermat
17th-century conjecture proved by Andrew Wiles in 1994
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that there are no positive integers a
Fermat's_Last_Theorem
Mathematics award
The Fermat Prize of mathematical research biennially rewards research works in fields where the contributions of Pierre de Fermat have been decisive:
Fermat_Prize
Every positive integer is a sum of at most n n-gonal numbers
In additive number theory, the Fermat polygonal number theorem states that every positive integer is a sum of at most n n-gonal numbers. That is, every
Fermat polygonal number theorem
Fermat_polygonal_number_theorem
Catalan number 1905 = Fermat pseudoprime 1907 = safe prime, balanced prime 1908 = coreful perfect number 1909 = hyperperfect number 1910 = number of compositions
1000_(number)
Fermat number Fermat point Fermat–Weber problem Fermat polygonal number theorem Fermat polynomial Fermat primality test Fermat pseudoprime Fermat quintic
List of things named after Pierre de Fermat
List_of_things_named_after_Pierre_de_Fermat
A prime p divides a^p–a for any integer a
In number theory, Fermat's little theorem states that if p is a prime number, then for any integer a, the number ap − a is an integer multiple of p. In
Fermat's_little_theorem
Prime number of the form 2^n – 1
r = 1, it is a Mersenne number. When p = 2, it is a Fermat number. The only known Mersenne–Fermat primes with r > 1 are MF(2, 2), MF(2, 3), MF(2, 4),
Mersenne_prime
Non-fiction book by Simon Singh
Fermat's Last Theorem is a popular science book (1997) by Simon Singh. It tells the story of the search for a proof of Fermat's Last Theorem, first conjectured
Fermat's_Last_Theorem_(book)
American mathematician (1927–2010)
them in a number of books and articles. Selfridge made the following conjecture about the Fermat numbers Fn = 22n + 1 . Let g(n) be the number of distinct
John_Selfridge
Two raised to an integer power
of two are common in computing. The first 21 of them are: Also see Fermat number, Tetration and Hyperoperation § Lower hyperoperations. All of these
Power_of_two
Primality test for Fermat numbers
test is a primality test, which can be used to determine whether a Fermat number is prime. It is a variant of Proth's test. The test is named after a
Pépin's_test
Natural number
262,468 = Leyland number 268,705 = Leyland number 271,129 – smallest known Sierpiński prime 274,177 = prime factor of the Fermat number F6 275,807/195,025
100,000
Natural number
, the first composite Fermat number. 4,294,968,320 : Leyland number using 2 and 32 (232 + 322) 4,295,032,832 : Leyland number using 4 and 16 (416 + 164)
1,000,000,000
Prime such that p^2 divides 2^(p-1)-1
In number theory, a Wieferich prime is a prime number p such that p2 divides 2p − 1 − 1, therefore connecting these primes with Fermat's little theorem
Wieferich_prime
Composite number that passes Fermat's probable primality test
In number theory, the Fermat pseudoprimes make up the most important class of pseudoprimes that come from Fermat's little theorem. Fermat's little theorem
Fermat_pseudoprime
1995 publication in mathematics
Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves
Wiles's proof of Fermat's Last Theorem
Wiles's_proof_of_Fermat's_Last_Theorem
Natural number
sphenic number, an octagonal number, a Smith number, a Harshad number and a Fermat pseudoprime to base 2. 646 = 2 × 17 × 19. It is a sphenic number There
600_(number)
Natural number
Fermat primes. In computing, 4,294,967,295 is the highest unsigned (that is, not negative) 32-bit integer, which makes it the highest possible number
4,294,967,295
Concept in Number Theory
In number theory, the Fermat quotient of an integer a with respect to an odd prime p is defined as q p ( a ) = a p − 1 − 1 p , {\displaystyle q_{p}(a)={\frac
Fermat_quotient
smallest base 2 Fermat pseudoprime. 496, the third perfect number. 1729, the Hardy–Ramanujan number, also known as the second taxicab number; that is, the
List_of_numbers
297 is a Fermat number and semiprime. It is the smallest number of the form 2 2 n + 1 {\displaystyle 2^{2^{n}}+1} which is not a prime number. Demographics
Orders_of_magnitude_(numbers)
Natural number
on their limbs. 5 is a Fermat prime, a Mersenne prime exponent, as well as a Fibonacci number. 5 is the first congruent number, as well as the length
5
Natural number
polygon. In number theory, primes of this form are known as Fermat primes, named after the mathematician Pierre de Fermat. The only known prime Fermat numbers
65,537
Branch of pure mathematics
simple to understand but are very difficult to solve. Examples of this are Fermat's Last Theorem, which was proved 358 years after the original formulation
Number_theory
Number divisible only by 1 and itself
de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also investigated the primality of the Fermat numbers
Prime_number
Algebraic curve
mathematics, the Fermat curve is the algebraic curve in the complex projective plane defined in homogeneous coordinates (X:Y:Z) by the Fermat equation: X n
Fermat_curve
Australian mathematician and computer scientist
factored the eighth Fermat number using a variant of the Pollard rho algorithm. He later factored the tenth and eleventh Fermat numbers using Lenstra's
Richard_P._Brent
Probabilistic primality test
The Fermat primality test is a probabilistic test to determine whether a number is a probable prime. Fermat's little theorem states that if p is prime
Fermat_primality_test
Natural number
71828... 3 is the first Mersenne prime. It is also the first of five known Fermat primes. It is the second Fibonacci prime (and the second Lucas prime), the
3
Number of form 2^(2^p-1)-1 with prime exponent
"Martian prime". Cunningham chain Double exponential function Fermat number Perfect number Wieferich prime Chris Caldwell, Mersenne Primes: History, Theorems
Double_Mersenne_number
Used to count, measure, and label
ISBN 978-3-031-83382-3. Deza, Elena (2021). Mersenne Numbers and Fermat Numbers. Selected Chapters Of Number Theory: Special Numbers. Vol. 1. World Scientific. pp
Number
Light rays follow quickest paths
Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. Fermat's principle states that the path
Fermat's_principle
Natural number
Journal of Number Theory. 161: 230–239. arXiv:1411.5354. doi:10.1016/j.jnt.2015.06.001. MR 3435726. S2CID 117748466. "Generalized Fermat primes in odd
71_(number)
Theorem on modular exponentiation
In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if n and a are coprime positive integers
Euler's_theorem
Index of articles associated with the same name
mathematician Pierre de Fermat engendered many theorems. Fermat's theorem may refer to one of the following theorems: Fermat's Last Theorem, about integer
Fermat's_theorem
Integer factorization algorithm
algorithm's most remarkable success was the 1980 factorization of the Fermat number F8 = 1238926361552897 × 93461639715357977769163558199606896584051237541638188580280321
Pollard's_rho_algorithm
British mathematician who proved Fermat's Last Theorem
Professor at the University of Oxford, specialising in number theory. He is best known for proving Fermat's Last Theorem, for which he was awarded the 2016 Abel
Andrew_Wiles
Number with prime Hamming weight
every number of the form 2 n + 1 {\displaystyle 2^{n}+1} with n > 1 {\displaystyle n>1} , including every Fermat number, is a pernicious number. This
Pernicious_number
Condition under which an odd prime is a sum of two squares
In additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed as: p = x 2 + y 2 , {\displaystyle p=x^{2}+y^{2}
Fermat's theorem on sums of two squares
Fermat's_theorem_on_sums_of_two_squares
Partial results found before the complete proof
Fermat's Last Theorem is a theorem in number theory, originally stated by Pierre de Fermat in 1637 and proven by Andrew Wiles in 1995. The statement of
Proof of Fermat's Last Theorem for specific exponents
Proof_of_Fermat's_Last_Theorem_for_specific_exponents
Generalisation of Fourier transform to any ring
of the number theoretic transform such as the Fermat Number Transform (m = 2k+1), used by the Schönhage–Strassen algorithm, or Mersenne Number Transform
Discrete Fourier transform over a ring
Discrete_Fourier_transform_over_a_ring
Prime number of the form k*(2^n)+1
be checked to deterministically verify or falsify the primality of a Fermat number. As of 2022[update], the largest known Proth prime is 10223 × 2 31172165
Proth_prime
Multiplication algorithm
makes N a Fermat number. When doing mod N = 2 M + 1 = 2 2 L + 1 {\displaystyle N=2^{M}+1=2^{2^{L}}+1} , we have a Fermat ring. Because some Fermat numbers
Schönhage–Strassen_algorithm
3rd-century Greek mathematician
The 1621 edition of Arithmetica by Bachet gained fame after Pierre de Fermat wrote his famous "Last Theorem" in the margins of his copy. In modern use
Diophantus
over-relaxation Chinese remainder theorem Fermat's little theorem Proofs of Fermat's little theorem Fermat quotient Euler's totient function Noncototient
List_of_number_theory_topics
Is every Fermat number 2 2 n + 1 {\displaystyle 2^{2^{n}}+1} composite for n > 4 {\displaystyle n>4} ? Is 509,203 the lowest Riesel number? The following
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
of proofs of Fermat's little theorem, which states that a p ≡ a ( mod p ) {\displaystyle a^{p}\equiv a{\pmod {p}}} for every prime number p and every integer
Proofs of Fermat's little theorem
Proofs_of_Fermat's_little_theorem
Island in the North Atlantic Ocean
Cosgrave was a specialist in number theory and discovered a 2000-digit prime number in 1999 and a record composite Fermat number in 2003. John Lighton Synge
Ireland
Natural number
four-variable pair is 1729. 1729 is the first number in the sequence of "Fermat near misses" defined, in reference to Fermat's Last Theorem, as numbers of the form
1729_(number)
Natural number
103 3 + 1 {\displaystyle 64^{3}+94^{3}=103^{3}+1} makes 103 part of a "Fermat near miss". There are 103 different connected series-parallel partial orders
103_(number)
Rational right triangles cannot have square area
Fermat's right triangle theorem is a non-existence proof in number theory, published in 1670 among the works of Pierre de Fermat, soon after his death
Fermat's right triangle theorem
Fermat's_right_triangle_theorem
Numbers obtained by adding the two previous ones
r=c{\sqrt {n}}} where n is the index number of the floret and c is a constant scaling factor; the florets thus lie on Fermat's spiral. The divergence angle,
Fibonacci_sequence
On the divisibility of solutions to Fermat's Last Theorem for prime exponent
solutions to the equation x p + y p = z p {\displaystyle x^{p}+y^{p}=z^{p}} of Fermat's Last Theorem for odd prime p {\displaystyle p} . Specifically, Sophie Germain
Sophie_Germain's_theorem
References to the famous problem in number theory
The problem in number theory known as "Fermat's Last Theorem" has repeatedly received attention in fiction and popular culture. It was proved by Andrew
Fermat's Last Theorem in fiction
Fermat's_Last_Theorem_in_fiction
Natural number
F_{11}(150)=150^{2048}+1} is the largest known generalized Fermat prime with even base less than 1000. 150 is also: The number of degrees in the quincunx astrological aspect
150_(number)
Generalization of Fermat's Last Theorem and of Catalan's conjecture,
In number theory, the Fermat–Catalan conjecture is a generalization of Fermat's Last Theorem and of Catalan's conjecture. The conjecture states that the
Fermat–Catalan_conjecture
Topics referred to by the same term
{\displaystyle \mathbb {F} _{4}} , the field with four elements F4, Fermat number F4 (classification), a wheelchair sport classification Formula 4, a
F4
Special-purpose integer factorization algorithm
Manasse, M. S. & Pollard, J. M. (1993), "The Factorization of the Ninth Fermat Number", Mathematics of Computation, 61 (203): 319–349, Bibcode:1993MaCom.
Special_number_field_sieve
Topics referred to by the same term
the United States Baseball League Pépin's test to determine whether a Fermat number is prime in mathematics Pippin (roller coaster), earlier version of
Pippin
Spiral that surrounds equal area per turn
A Fermat's spiral or parabolic spiral is a plane curve with the property that the area between any two consecutive full turns around the spiral is invariant
Fermat's_spiral
Natural number
Michal; Luca, Florian; Somer, Lawrence (2001). 17 Lectures on Fermat Numbers: From Number Theory to Geometry. CMS Books in Mathematics/Ouvrages de Mathématiques
1105_(number)
Factorization method based on the difference of two squares
Fermat's factorization method, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares: N = a
Fermat's_factorization_method
Natural number with a decimal representation made of repeated instances of the same digit
19, 23, 29, 37, 41, 47, 53, ... (sequence A220627 in the OEIS) If a Fermat number F n = 2 2 n + 1 {\displaystyle F_{n}=2^{2^{n}}+1} is prime, it is not
Repdigit
Fourier-related mathematical transform
nk}{N}}+{\frac {2\pi ml}{M}}).} At this point we present the Fermat number transform (FNT). The tth Fermat number is given by F t = 2 b + 1 {\displaystyle F_{t}=2^{b}+1}
Discrete_Hartley_transform
Probable prime that is composite
pseudoprimes. An integer x that is a Fermat pseudoprime to all values of a that are coprime to x is called a Carmichael number. Catalan pseudoprime Elliptic
Pseudoprime
Natural number
order 10 on the Mandelbrot set. Since 51 is the product of the distinct Fermat primes 3 and 17, a regular polygon with 51 sides is constructible with compass
51_(number)
Number used for counting
natural-number results: subtracting a larger natural number from a smaller one results in a negative number and dividing one natural number by another
Natural_number
Regular polygon that can be constructed with compass and straightedge
integer. A Fermat prime is a prime number of the form 2 ( 2 m ) + 1 {\displaystyle 2^{(2^{m})}+1} , where m ≥ 0 is an integer. The number of Fermat primes
Constructible_polygon
Natural number
- 43). 17 is one of five known Fermat primes, and one of six total lucky numbers of Euler. Since seventeen is a Fermat prime, regular heptadecagons can
17_(number)
Natural number
progression (251 + 257 + 263). Since 771 is the product of the distinct Fermat primes 3 and 257, a regular polygon with 771 sides can be constructed using
700_(number)
Branch of number theory
In this book Gauss brings together results in number theory obtained by mathematicians such as Fermat, Euler, Lagrange and Legendre and adds important
Algebraic_number_theory
Conjecture in number theory
in part due to its large number of consequences in number theory including Roth's theorem, Faltings' theorem, the Fermat–Catalan conjecture, and Brocard's
Szpiro's_conjecture
Danish mathematician and astronomer (1801–1885)
possible solutions to this problem. In 1854, he factored the sixth Fermat number as 264+1 = 67280421310721 × 274177. Hockey, Thomas (2009). The Biographical
Thomas Clausen (mathematician)
Thomas_Clausen_(mathematician)
Integer having a non-trivial divisor
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly, it is a positive integer that has
Composite_number
Natural number
341 is the smallest Fermat pseudoprime; it is the least composite odd modulus m greater than the base b, that satisfies the Fermat property "bm−1 − 1 is
300_(number)
Integer factorization algorithm
Ninth Fermat Number, Math. Comp. 61 (1993), 319-349. Available at [2]. A. K. Lenstra, H. W. Lenstra, Jr. (eds.) The Development of the Number Field Sieve
Rational_sieve
(Mathematical) decomposition into a product
inefficient for larger integers. For example, Pierre de Fermat was unable to discover that the 6th Fermat number 1 + 2 2 5 = 1 + 2 32 = 4 294 967 297 {\displaystyle
Factorization
Prime number of the form 2^u × 3^v + 1
therefore a Fermat prime (unless u = 0). If v is positive then u must also be positive (because 3 v + 1 {\displaystyle 3^{v}+1} would be an even number greater
Pierpont_prime
Result concerning properties of Galois representations associated with modular forms
proven by Ken Ribet. The proof was a significant step towards the proof of Fermat's Last Theorem (FLT). As shown by Serre and Ribet, the Taniyama–Shimura conjecture
Ribet's_theorem
Dutch mathematician (born 1956)
of the number field sieve. With coauthors, he showed the great potential of the algorithm early on by using it to factor the ninth Fermat number, which
Arjen_Lenstra
Polynomial equation whose integer solutions are sought
testing if a rational number is the dth power of another rational number). A witness of the difficulty of the problem is Fermat's Last Theorem (for d >
Diophantine_equation
Computer algebra system
Fermat (named after Pierre de Fermat) is a computer algebra system developed by Prof. Robert H. Lewis of Fordham University. It can work on integers (of
Fermat (computer algebra system)
Fermat_(computer_algebra_system)
Natural number
{\displaystyle 2^{2^{n}}+1,} specifically with n = 3, and therefore a Fermat prime. Thus, a regular polygon with 257 sides is constructible with compass
257_(number)
Numbers with a certain property involving recursive summation
In number theory, a happy number is a number which eventually reaches 1 when the number is replaced by the sum of the square of each digit. For instance
Happy_number
Number equal to the sum of its proper divisors
( 2 n + 1 ) {\displaystyle 2^{n-1}(2^{n}+1)} formed as the product of a Fermat prime 2 n + 1 {\displaystyle 2^{n}+1} with a power of two in a similar way
Perfect_number
French engineer (born 1931)
Development: 498–504. Nussbaumer, Henri J. (1976). "Complex Convolutions via Fermat Number Transforms". IBM Journal of Research and Development: 282–284. Nussbaumer
Henri_Nussbaumer
Mathematical project in integer factorization
n ≥ 2; these are the generalized Fermat numbers, which are Fermat numbers when b = 2. Any factor of a Fermat number 22n + 1 is of the form k·2n+2 + 1
Cunningham_Project
Bulgarian mathematician (born 1986)
2022. In 2025, he received the Salem Prize and the Fermat Prize, and in 2026, the Cole Prize in Number Theory. In 2019 Dimitrov proved the Schinzel–Zassenhaus
Vesselin_Dimitrov
Natural number
form the first twin-prime pair. Because 15 is the product of distinct Fermat primes, 3 and 5, a regular pentadecagon is constructible with a compass
15_(number)
Natural number
natural number following 216 and preceding 218. 217 is a centered hexagonal number, a 12-gonal number, a centered 36-gonal number, a Fermat pseudoprime
217_(number)
Prime pair of the form (p, 2p+1)
her investigations of Fermat's Last Theorem. One attempt by Germain to prove Fermat's Last Theorem was to let p be a prime number of the form 8k + 7 and
Safe and Sophie Germain primes
Safe_and_Sophie_Germain_primes
Natural number
Brown number pairs. 120 appears in Pierre de Fermat's modified Diophantine problem as the largest known integer of the sequence 1, 3, 8, 120. Fermat wanted
120_(number)
Area of a right triangle with rational-numbered sides
right triangle theorem, named after Pierre de Fermat, states that no square number can be a congruent number. However, in the form that every congruum (the
Congruent_number
expressions in modular arithmetic which, if they hold, entail the truth of Fermat's Last Theorem. Since the theorem has now been proven, these are now of mainly
Mirimanoff's_congruence
Composite number in number theory
numbers with the "Fermat property", or "F numbers" for short. Fermat's little theorem states that if p {\displaystyle p} is a prime number, then for any integer
Carmichael_number
Integer having only small prime factors
In number theory, an n-smooth (or n-friable) number is an integer whose prime factors are all less than or equal to n. For example, a 7-smooth number is
Smooth_number
Probabilistic primality testing algorithm
combination of a strong Fermat probable prime test to base 2 and a standard or strong Lucas probable prime test. The Fermat and Lucas test each have
Baillie–PSW_primality_test
Natural number
{\displaystyle F_{7}(113)={\frac {113^{128}+1}{2}}} is the smallest generalized Fermat prime with odd base and k ≥ 7 {\displaystyle k\geq 7} . "Sloane's A005384 :
113_(number)
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