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Fermat number

A Fermat number is a positive integer of the form F_n = 2^{2^n} + 1, where n is a non-negative integer. These numbers are named after the French mathematician Pierre de Fermat (1601–1665), who first studied them and conjectured that all such numbers are prime. The first five Fermat numbers—for n = 0 to $4—are indeed prime: F_0 = 3, F_1 = 5, F_2 = 17, F_3 = 257, and F_4 = 65537. These are the only known Fermat primes, and all Fermat numbers for $5 \leq n \leq 32 are known to be composite, with F_5 = 4{,}294{,}967{,}297 = 641 \times 6{,}700{,}417 being the first counterexample discovered by Leonhard Euler in 1732. It remains an open question whether there are any additional prime Fermat numbers beyond the first five. Fermat numbers possess several notable properties, including that they are pairwise coprime: for distinct non-negative integers m and n, \gcd(F_m, F_n) = 1. Additionally, the product of the first n Fermat numbers plus 2 equals the next one: F_n = F_0 F_1 \cdots F_{n-1} + 2 for n \geq 1. They also have applications in ; proved that a regular m-gon is constructible with and if and only if m = 2^k p_1 p_2 \cdots p_t, where the p_i are distinct Fermat primes.

Introduction

Definition

A Fermat number is defined as F_n = 2^{2^n} + 1, where n is a . The first five Fermat numbers are F_0 = 3, F_1 = 5, F_2 = 17, F_3 = 257, and F_4 = 65537. introduced these numbers in a 1640 letter to Bernard Frénicle de Bessy while discussing methods for constructing perfect numbers. In that correspondence, Fermat noted their form and suggested they might all be prime, though he provided no proof. Fermat numbers grow extremely rapidly in magnitude; for example, F_n consists of exactly $2^n + 1 digits, reflecting the tower in the exponent.

Historical Context

In 1640, introduced the form of numbers now known as Fermat numbers in a letter to Bernard Frénicle de Bessy, without proof that all such numbers are prime based on his verification of the first few cases. This stemmed from Fermat's broader explorations in , where he observed the primality of the initial instances and posited it as a general property. Leonhard Euler verified the primality of the first five Fermat numbers, F_0 through F_4, in 1732, but disproved Fermat's conjecture by factoring F_5 = 4{,}294{,}967{,}297 = 641 \times 6{,}700{,}417, demonstrating that F_5 is composite. Euler's factorization relied on properties of factors of Fermat numbers, specifically identifying 641 as a divisor (of the form k \cdot 128 + 1) through systematic checking, and confirming the primality of the cofactor 6,700,417 by trial division up to its square root. This discovery marked the first known composite Fermat number and shifted early interest from universal primality to the structure and factors of these numbers. It remains unknown whether there are any Fermat primes beyond the first five. During the 19th century, interest in Fermat numbers persisted and culminated in Théophile Pépin's development of a primality test in 1877 specifically for Fermat numbers, providing a deterministic method using quadratic reciprocity to check if $3^{(F_n-1)/2} \equiv -1 \pmod{F_n}. Pépin's test built on earlier ideas from Lucas and offered an efficient way to verify primality for larger Fermat numbers without full factorization. Meanwhile, factorization efforts advanced, with Édouard Lucas and others exploring algebraic properties, though progress on higher composites remained limited until computational tools emerged. In the 20th century, computational advances enabled further factorizations, such as Michael A. Morrison and John Brillhart's 1975 complete factorization of F_7 using the continued fraction method, revealing it as a product of five prime factors and highlighting the increasing difficulty of these tasks. This work exemplified the evolution of factoring algorithms tailored to Fermat numbers' special form. Ongoing projects, like PrimeGrid, have continued this trend into the ; for instance, in early 2025, PrimeGrid discovered a 2,397,178-digit prime factor of a high-index Fermat number, underscoring the scale of modern efforts to probe their compositeness. Post-Euler, the focus shifted decisively from expecting all Fermat numbers to be prime toward investigating their composite nature, algebraic factorizations, and generalizations in .

Mathematical Properties

Basic Algebraic Properties

Fermat numbers possess several fundamental algebraic properties that distinguish them from other sequences of integers. A key characteristic is their pairwise coprimality: for any distinct nonnegative integers m and n, \gcd(F_m, F_n) = 1. This property implies that no prime of one Fermat number can divide another, ensuring that the prime factors of different Fermat numbers are entirely disjoint. The proof of this coprimality, originally established by in 1730, relies on showing that F_n = \prod_{i=0}^{n-1} F_i + 2 for n \geq 1, from which it follows that any common of F_m and F_n (with m < n) must divide 2, but since all Fermat numbers are odd, the gcd is 1. In binary representation, each Fermat number F_n = 2^{2^n} + 1 appears as a 1 followed by $2^n - 1 zeros followed by another 1. For example, F_0 = 3 = 11_2 (no zeros), F_1 = 5 = 101_2 (one zero), and F_2 = 17 = 10001_2 (three zeros). This structure results in a binary length of exactly $2^n + 1 bits, with precisely two 1's and the remainder zeros, underscoring their sparse binary form. This binary form directly connects Fermat numbers to powers of 2, as F_n - 2 = 2^{2^n}, revealing that they differ from a pure power of 2 by only 1 in the units place. This near-power-of-2 nature facilitates certain computational analyses but also contributes to their challenging factorization for larger n. The magnitude of Fermat numbers grows double-exponentially, making them extremely large even for modest n. The number of decimal digits in F_n is \lfloor \log_{10} F_n \rfloor + 1 \approx 2^n \log_{10} 2 + 1 \approx 0.3010 \times 2^n + 1, highlighting their rapid increase; for instance, F_9 has 155 digits. This explosive growth poses significant obstacles for primality testing and factorization beyond small indices.

Recurrence and Product Formulas

Fermat numbers satisfy the fundamental product identity \prod_{k=0}^{m-1} F_k = F_m - 2 for every integer m \geq 1. This relation, first established by in a 1730 letter to , arises from the geometric series factorization $2^{2^m} - 1 = \prod_{k=0}^{m-1} (2^{2^k} + 1), since F_m = 2^{2^m} + 1 implies F_m - 2 = 2^{2^m} - 1. From the definition F_n = 2^{2^n} + 1, a basic recurrence follows directly: F_{n+1} = 2^{2^{n+1}} + 1 = (2^{2^n})^2 + 1 = (F_n - 1)^2 + 1 = F_n^2 - 2 F_n + 2. A more general recurrence relating arbitrary indices is F_{m+n} = (F_m - 1)^{2^n} + 1, which generalizes the basic case by setting x = 2^{2^m} = F_m - 1, so F_{m+n} = x^{2^n} + 1. These recurrences highlight the exponential growth and nested structure inherent to the sequence. The product identity has key consequences for the prime factors of Fermat numbers. Specifically, it implies that distinct Fermat numbers are pairwise coprime. To see this, suppose a prime p divides both F_m and F_n with m > n. Then p divides F_m - 2 = \prod_{k=0}^{m-1} F_k, so p divides some F_k for $0 \leq k < m; in particular, since p divides F_n, but all Fermat numbers exceed 2 and are odd, p cannot divide 2, leading to a contradiction unless no such p exists. Certain composite Fermat numbers admit special algebraic factorizations akin to Aurifeuillean identities, which exploit polynomial decompositions of forms like x^{2^{r}} + 1 for specific exponents. These factorizations, while not applying universally to the sequence, provide explicit non-trivial decompositions for individual terms beyond the general product relations.

Primality Status

Known Fermat Primes

The five known Fermat primes are F_0 = 3, F_1 = 5, F_2 = 17, F_3 = 257, and F_4 = 65537. These were verified as prime by Leonhard Euler in 1732 using trial division to check for divisors up to the square root of each number. For F_0 through F_3, the primality follows directly from their small values and absence of prime factors less than their square roots, which Euler confirmed via exhaustive checks. The case of F_4 = 65537, the largest known , required testing divisibility by all primes up to 251 (since \sqrt{65537} \approx 256); Euler adapted methods akin to early primality tests, such as checking potential factors of the form k \cdot 2^{n+2} + 1, to establish its primality without finding any divisors. No Fermat primes beyond F_4 are known as of 2025, with all F_n for $5 \leq n \leq 32 proven composite through direct or discovery of nontrivial factors. These results stem from extensive computational efforts, including the Cunningham project, which have ruled out primality for these cases via explicit factorizations for smaller n and partial factorizations or primality certificates for larger ones up to n=32.

Composite Fermat Numbers and Factorizations

The fifth Fermat number, F_5 = 2^{32} + 1 = 4,294,967,297, is the first known composite in the sequence and factors as $641 \times 6,700,417, where both factors are prime; this factorization was discovered by in 1732. Euler's result disproved Fermat's conjecture that all Fermat numbers are prime, and the factors conform to the general form k \cdot 2^{n+2} + 1 expected for divisors of F_n. Subsequent Fermat numbers have also proven composite, with complete factorizations known up to F_{11}. For instance, F_6 = 2^{64} + 1 = 18,446,744,073,709,551,617 = 274,177 \times 67,280,421,310,721, both primes, factored by Clausen and Landry in 1855. The seventh, F_7 = 2^{128} + 1 = 340,282,366,920,938,463,463,374,607,431,768,211,457 = 59,649,589,127,497,217 \times 5,704,689,200,685,129,054,721, both primes, was factored using the continued fraction method by Michael Morrison and John Brillhart in 1970–1975. F_8 factors into two primes (discovered by Richard Brent and John Pollard in 1980), F_9 into three primes (1903–1990), F_{10} into four primes (1953–1995), and F_{11} into five primes (1899–1988). The following table summarizes the complete factorizations of F_5 through F_{11}, highlighting the smallest prime factor for each:
Fermat NumberSmallest Prime FactorNumber of Prime FactorsDiscovery Notes
F_56412Euler, 1732
F_6274,1772Clausen/Landry, 1855
F_759,649,589,127,497,2172Morrison/Brillhart, 1970–1975
F_81,238,926,361,552,8972Brent/Pollard, 1980
F_92,424,8333Multiple, 1903–1990
F_{10}45,592,5774Multiple, 1953–1995
F_{11}319,4895Multiple, 1899–1988
All factors across these numbers are of the form k \cdot 2^{n+2} + 1, a structural property that guides factorization searches. As of November 2025, F_{12} through F_{32} are known to be composite but only partially factored, with multiple prime factors identified for each yet composite cofactors remaining. For example, F_{12} has six known prime factors, F_{13} has four, and F_{15} has three, all with unfactored composite remainders. Distributed computing efforts, such as those by , continue to uncover new factors; a notable recent discovery is the factor $99 \cdot 2^{5,798,449} + 1 of F_{5,798,447} on January 14, 2025, by Valter Cavecchia and PrimeGrid volunteers. For n > 32, factorization searches have confirmed compositeness for many via isolated factors, while others await further testing due to their immense size, with no complete primality verifications possible yet.

Testing Compositeness

Heuristic Arguments

Heuristic arguments for the compositeness of Fermat numbers rely on probabilistic models derived from the , which estimates the likelihood that a random of size approximately x is prime as roughly $1 / \ln x. For the Fermat number F_n = 2^{2^n} + 1, \ln F_n \approx 2^n \ln 2, so the probability that F_n is prime is approximately $1 / (2^n \ln 2). This probability decreases double-exponentially with n, making large Fermat numbers overwhelmingly likely to be composite. The expected total number of Fermat primes is the infinite sum \sum_{n=0}^\infty 1 / (2^n \ln 2) = 2 / \ln 2 \approx 2.885. This finite value indicates that only finitely many Fermat primes are anticipated overall, aligning with the observation of exactly five known primes and suggesting no additional ones are expected. reinforces this heuristic: all Fermat numbers F_n for $5 \leq n \leq 32 have been shown to be composite, and as of November 2025, F_{33} remains of unknown primality. The special algebraic form $2^{2^n} + 1 contributes to this pattern, as it predisposes larger Fermat numbers to having algebraic factors, increasing the likelihood of compositeness beyond what a random model alone would predict.

Equivalent Conditions and Deterministic Tests

A fundamental deterministic for Fermat numbers F_n = 2^{2^n} + 1 with n \geq 1 is Pépin's test, which asserts that F_n is prime $3^{(F_n - 1)/2} \equiv -1 \pmod{F_n}. This condition leverages the fact that 3 is a non-residue modulo F_n when F_n is prime, allowing a single to conclusively determine primality. The test has been computationally verified for the known Fermat primes F_0 through F_4, where the congruence holds, and fails for composite cases, establishing their status efficiently. Pépin's test is a special case of Proth's theorem, which provides a general criterion for primes of the form N = k \cdot 2^m + 1 where k is odd and k < 2^m. For Fermat numbers, k = 1 and m = 2^n, satisfying the conditions since $1 < 2^{2^n}. Proth's theorem states that such an N is prime there exists a quadratic non-residue a modulo N such that a^{(N-1)/2} \equiv -1 \pmod{N}. In the Fermat case, a = 3 serves as this non-residue, directly yielding Pépin's test and enabling deterministic verification without needing to find factors. An adaptation of the Lucas-Lehmer test, originally for Mersenne numbers, provides an equivalent sequence-based criterion for Fermat numbers. More generally, a Lucas-Lehmer-like test states that F_n (for n > 1) is prime if and only if F_n divides S_{2^n - 2}, where S_0 = 5 and S_i = S_{i-1}^2 - 2 for i \geq 1. This sequence method is equivalent to Pépin's test through properties of Lucas sequences, offering an alternative computational path that aligns with the in Pépin for higher n. In modern computations, deterministic variants of the Miller-Rabin test, using specific witness sets tailored to the size and form of Fermat numbers, complement these classical tests for confirming compositeness. For instance, failure of the Miller-Rabin condition with bases like 2, 3, and 5 establishes compositeness without factorization. These methods, alongside Pépin's test failures, have verified the compositeness of F_5 through F_{32} as of 2025, often followed by partial factorizations using elliptic curve methods or the special number field sieve. For example, F_{20} and F_{24} were shown composite via Pépin's test without initial factors, while others like F_5 to F_{11} are fully factored.

Advanced Theorems

Zsigmondy's Theorem and Primitive Prime Factors

Zsigmondy's theorem provides a powerful result on the existence of primitive prime divisors in sequences of the form a^n \pm b^n. Specifically, for integers a > b \geq 1 with \gcd(a, b) = 1 and n \geq 2, the number a^n + b^n has at least one primitive prime divisor—a prime p that divides a^n + b^n but does not divide a^k + b^k for any $1 \leq k < n—with the sole exception being (a, b, n) = (2, 1, 3), where $2^3 + 1^3 = 9 = 3^2 has no such prime. This theorem, originally proved by in 1892, guarantees new prime factors at each step in such sequences beyond the exceptional cases. Fermat numbers F_n = 2^{2^n} + 1 arise naturally in this context through their expression in terms of cyclotomic polynomials: F_n = \Phi_{2^{n+1}}(2), where \Phi_m(x) is the m-th cyclotomic polynomial. Zsigmondy's theorem applies directly here by considering the sequence a_k = 2^{2^k} + 1 with a = 2, b = 1, and exponents that are powers of 2 (even for k \geq 1). Since the exponent $2^n \geq 2 avoids the sole exception for the + case (which requires an odd exponent 3), each F_n for n \geq 0 possesses at least one primitive prime divisor p_n, meaning p_n divides F_n but no F_k for k < n. The pairwise coprimality of Fermat numbers further ensures that all prime factors of F_n are primitive in this sense. Such a primitive prime divisor p_n of F_n satisfies p_n \equiv 1 \pmod{2^{n+2}} for n \geq 2, as the multiplicative order of 2 modulo p_n is exactly $2^{n+1}, but a refinement using quadratic residues shows the stronger congruence. For n = 0 and n = 1, the congruences hold modulo $2^{n+1}. This property follows from $2^{2^n} \equiv -1 \pmod{p_n}, implying the order divides $2^{n+1} but not smaller powers of 2. There are no exceptions to the existence of these primitive primes for any known Fermat number. The existence of distinct primitive prime divisors p_n for each n implies that there are infinitely many primes of the form k \cdot 2^{n+2} + 1 (for varying n \geq 2), as the p_n are distinct due to the pairwise coprimality of the F_n. This provides a concrete infinite family of primes with prescribed form in the cyclotomic field extensions generated by roots of unity of order $2^{n+2}.

Other Theorems on Divisibility and Structure

The lifting the exponent lemma provides bounds on the p-adic valuation v_p(F_n) for odd primes p dividing a Fermat number F_n = 2^{2^n} + 1. Specifically, if an odd prime p divides F_n, then v_p(F_n) = 1 unless p satisfies the stronger condition $2^{p-1} \equiv 1 \pmod{p^2}, meaning p is a . In the latter case, the valuation could exceed 1, but computations show that the only known Wieferich primes to base 2—1093 and 3511—do not divide any Fermat number up to F_{32}. Consequently, all completely factored Fermat numbers are squarefree, with no squared prime factors. Sierpiński's theorem establishes the existence of infinitely many odd positive integers k, called Sierpiński numbers, such that k \cdot 2^n + 1 is composite for every natural number n \geq 1. The original proof constructs such k using a covering system modulo the product of small primes derived from factors of the first few (e.g., 3, 5, 17, 257, 65537), ensuring that for every n, k \cdot 2^n + 1 is divisible by one of these primes under the covering congruences. This result highlights the role of Fermat number factors in demonstrating systematic compositeness for certain linear forms in exponents, with the smallest such k being 78557. The product formula F_n - 2 = \prod_{i=0}^{n-1} F_i implies profound structural properties for divisibility among Fermat numbers. In particular, it proves that distinct Fermat numbers are pairwise coprime, as any common odd prime divisor p of F_m and F_n (with m < n) would divide F_n - 2, hence divide 2, a contradiction. This ensures that prime factors of composite Fermat numbers like F_5 (divided by 641) are unique to that number and do not divide higher F_n. While the compositeness of a lower F_m does not force higher F_n to be composite, the formula facilitates algebraic factorizations and confirms that higher F_n inherit no shared divisors, though their own compositeness is verified through direct searches for factors of the form k \cdot 2^{n+2} + 1. Fermat numbers F_n = 2^{2^n} + 1 and M_p = 2^p - 1 (for prime p) share structural analogies as repunit-like forms in base 2, both conjectured (incorrectly for Fermat) to yield infinitely many primes and used historically in primality pursuits. Prime factors of both exhibit similar congruence restrictions: divisors of M_p are of the form k \cdot 2p + 1, while divisors of F_n (for n \geq 1) are of the form k \cdot 2^{n+1} + 1, reflecting the multiplicative order of 2 modulo the prime. However, their divisibility patterns diverge markedly—Mersenne numbers may share factors across different p if orders align, whereas Fermat numbers' pairwise coprimality (from the product formula) guarantees distinct factors, making Fermat factorizations more isolated but computationally intensive due to the double-exponential growth. These parallels extend to their roles in , where both are evaluations of \Phi_{2^{m}}(2) for appropriate m.

Geometric Applications

Relation to Constructible Polygons

The Gauss–Wantzel theorem establishes that a regular N-gon is constructible using only a compass and straightedge if and only if N = 2^k \prod_{i=1}^t p_i, where k \geq 0 is an integer and the p_i are distinct . Fermat primes play a crucial role in this criterion because they are the only odd primes p for which the p-th has degree \phi(p) = p-1 = 2^m over the rationals for some integer m, enabling the embedding into a tower of quadratic extensions necessary for compass-and-straightedge constructions. The five known Fermat primes—F_0 = 3, F_1 = 5, F_2 = 17, F_3 = 257, and F_4 = 65537—permit the construction of regular N-gons where the odd part of N is any product of a subset of these primes multiplied by a power of 2. For instance, the regular pentagon arises from N = 5 = F_1. Since no additional Fermat primes are known and all tested higher Fermat numbers F_n for n \geq 5 are composite, the variety of constructible regular polygons is restricted by the limited supply of these primes. The largest known constructible regular polygon incorporating all five Fermat primes in its odd part has $3 \times 5 \times 17 \times 257 \times 65537 = 2^{32} - 1 = 4{,}294{,}967{,}295 sides.

Implications for Compass-and-Straightedge Constructions

The presence of Fermat primes significantly influences compass-and-straightedge constructions by enabling the generation of number fields that align with the quadratic nature of these tools. Specifically, for a Fermat prime F_n = 2^{2^n} + 1, the field \mathbb{Q}(\cos(2\pi / F_n)) has degree $2^{2^n - 1} over \mathbb{Q}, a power of 2 that permits construction through a tower of quadratic extensions. This structure allows the explicit construction of coordinates and lengths within these fields, expanding the set of achievable geometric figures beyond basic rationals and square roots. The compositeness of all known Fermat numbers beyond F_4 = 65537 imposes fundamental limits on these constructions, as no additional Fermat primes are available to extend the tower further for n > 4. This halts the discovery of new "Fermat-type" cyclotomic fields of pure power-of-2 degree, constraining the solvable higher-degree equations and the complexity of constructible figures to combinations of the five known primes (3, 5, 17, 257, ). Consequently, classical methods cannot access certain geometric solutions that would require infinite or unknown Fermat primes. In contemporary applications, the known Fermat primes underpin computer-assisted geometry proofs, where algorithms simulate successive quadratic extensions to verify constructibility for intricate figures, aiding research in algebraic geometry and educational tools.

Computational and Number-Theoretic Applications

Role in Pseudoprimes and Primality Testing

The Fermat primality test is a probabilistic method for assessing whether a number n > 1 is prime, based on Fermat's Little Theorem, which states that if p is prime and a is an integer not divisible by p, then a^{p-1} \equiv 1 \pmod{p}. To apply the test, select a base a coprime to n and compute a^{n-1} \pmod{n}; if the result is not 1, then n is composite, but if it is 1, n may be prime or a pseudoprime to base a. A Fermat pseudoprime to base a is a composite number that satisfies this congruence despite not being prime. Fermat numbers, defined as F_m = 2^{2^m} + 1, exhibit a special property with respect to base 2: $2^{2^m} \equiv -1 \pmod{F_m}, which implies $2^{2^{m+1}} = (2^{2^m})^2 \equiv 1 \pmod{F_m}. Since F_m - 1 = 2^{2^m}, it follows that $2^{F_m - 1} = 2^{2^{2^m}} = (2^{2^m})^{2^m} \equiv (-1)^{2^m} \equiv 1 \pmod{F_m}. Thus, every Fermat number passes the Fermat test to base 2, and every composite Fermat number is a base-2 Fermat pseudoprime. More strongly, every composite Fermat number is a strong pseudoprime to base $2^t for every integer t \geq 1, meaning it satisfies the conditions of the Miller-Rabin primality test for these bases. A prominent example is the fifth Fermat number F_5 = 2^{32} + 1 = 4,294,967,297, which Leonhard Euler factored in as $641 \times 6,700,417, proving it composite and the first known to Fermat's that all Fermat numbers are prime. As a composite Fermat number, F_5 is a base-2 and appears in test suites for primality algorithms to verify detection of known composites. In the broader context of primality testing, the form of Fermat numbers—with n-1 a high power of 2—highlights challenges in probabilistic tests like Miller-Rabin, which strengthens the Fermat test by writing n-1 = 2^s d with d odd and checking intermediate powers for \pm 1. Composite Fermat numbers possess unusually many bases due to their algebraic structure, informing the selection of witness bases in deterministic variants of Miller-Rabin to ensure composites like these are detected with multiple trials. This property underscores the need for diverse bases in practice, as relying solely on base 2 would fail to identify such pseudoprimes.

Generalizations

Fermat Numbers of Higher Orders

Higher-order Fermat numbers generalize the classical Fermat numbers by allowing the base to vary beyond 2. Defined as F_n(k) = k^{2^n} + 1 for integers k > 1 and nonnegative integers n, these numbers reduce to the standard Fermat numbers when k = 2. They share key structural properties with their classical counterparts, including pairwise coprimality for fixed k, meaning \gcd(F_m(k), F_n(k)) = 1 for m \neq n. Additionally, any odd prime divisor p of F_n(k) must be of the form p = c \cdot 2^{r} + 1 where r > n and c is odd. A fundamental property is the product formula \prod_{i=0}^{m-1} F_i(k) = \frac{F_m(k) - 2}{k - 1} for m \geq 1, which follows from the algebraic k^{2^m} - 1 = (k - 1) \prod_{i=0}^{m-1} F_i(k) and substituting the definition of F_m(k). This identity highlights the recursive structure and divisibility relations among the terms for fixed k, analogous to the classical case where it simplifies to \prod_{i=0}^{m-1} F_i(2) = F_m(2) - 2. These properties facilitate efforts and theoretical analysis. For specific bases k > 2, the numbers are typically composite for small n, with primes being rare. For example, with k = 3, F_0(3) = 4 = 2^2, F_1(3) = 10 = 2 \cdot 5, and F_2(3) = 82 = 2 \cdot 41 are all composite. Primes do occur sporadically, such as $6^2 + 1 = 37 and $10^2 + 1 = [101](/page/101) for n = 1, but they become increasingly scarce as n grows, with the largest known examples involving enormous exponents. A notable recent is the first prime of the form k^{2^{21}} + 1 found in October 2025, exceeding 13 million digits. These higher-order Fermat numbers are extensively studied in computational number theory, particularly through the Cunningham Project, which maintains factorization tables for forms like k^{2^n} + 1 to identify prime factors and explore their distribution.

Generalized Fermat Numbers of the Form F_n(a)

Generalized Fermat numbers of the form F_n(a) = a^{2^n} + 1, where a > 2 is an even positive integer and n \geq 0, extend the classical Fermat numbers by varying the base a while preserving the exponent as a power of 2. These numbers are of interest in number theory due to their structural similarities to Fermat numbers and their potential for primality, though they can only be prime when a is even, as odd a > 1 yields even values greater than 2, hence composite. When a = 2, F_n(2) recovers the original Fermat numbers F_n. Primality searches for F_n(a) have identified several primes, particularly for small n. For instance, F_5(6) = 6^{32} + 1 is a known prime, discovered in the 1980s as part of early computational efforts. These searches often leverage the form's alignment with , where F_n(a) fits the structure k \cdot 2^m + 1 for appropriate k and m, enabling efficient testing. Properties akin to Pépin's test for classical Fermat numbers generalize here via : F_n(a) is prime if a suitable quadratic non-residue base raised to (F_n(a) - 1)/2 yields -1 modulo F_n(a). Factorization of composite F_n(a) presents significant challenges due to their large size, but algebraic methods provide breakthroughs for specific bases. Aurifeuillian identities allow non-trivial factorizations for certain a, such as when a = 14, where splittings occur for particular n based on cyclotomic polynomial decompositions. For example, expressions like $14^{2^n} + 1 exhibit known algebraic factors derived from these identities, aiding partial resolutions in projects like the Cunningham tables. Such techniques highlight the interplay between algebraic structure and computational number theory in studying these numbers.

Bivariate Generalized Fermat Numbers F_n(a, b)

Bivariate generalized Fermat numbers are defined as F_n(a, b) = a^{2^n} + b^{2^n}, where a and b are positive with a > b > 0 and typically assumed coprime, and n is a non-negative . This form encompasses the univariate generalized Fermat numbers as the special case F_n(a, 1) = a^{2^n} + 1, and further reduces to the classical Fermat numbers when a = 2 and b = 1. The expression is homogeneous of degree $2^n, satisfying F_n(ka, kb) = k^{2^n} F_n(a, b) for any positive integer k. A fundamental divisibility property mirrors the product theorem for classical Fermat numbers: \prod_{k=0}^{n-1} F_k(a, b) = \frac{F_n(a, b) - 2 b^{2^n}}{a - b}. This identity implies that each earlier term F_m(a, b) for m < n divides F_n(a, b) - 2 b^{2^n}, facilitating proofs of coprimality among distinct terms under coprimality assumptions on a and b. Such numbers are prime only in rare cases. For example, F_1(2, 1) = 2^2 + 1^2 = 5 and F_2(3, 2) = 3^4 + 2^4 = 97 are both prime. Primes of this form require even a in many instances, and their scarcity increases with n and larger bases. These numbers appear in the study of , where their prime factors inform the structure of curves over finite fields, and in Diophantine equations, particularly generalized Fermat equations of the form x^p + y^q = z^r, aiding in bounding solutions via modular methods. Factorization techniques, such as those leveraging differences of powers or aurifeuillian identities for specific a and b, exploit the algebraic structure to decompose composite instances.

Largest Known Generalized Fermat Primes

Generalized Fermat primes, of the form a^{2^n} + 1 where a > 1 and both a and n are positive integers, continue to be subjects of active computational searches, primarily through distributed projects like PrimeGrid's Generalized Fermat Prime Search (GFNS). These efforts focus on varying a (often called k) for fixed large exponents n, as higher n yield numbers with exponentially increasing digits, making exhaustive searches infeasible without massive computational resources. As of November 2025, the largest known such prime is $2524190^{2^{21}} + 1, a 13,426,224-digit number discovered by volunteer Tom Greer on October 12, 2025, using PrimeGrid's GFNS subproject. This prime, denoted as GFN-21 for base k = 2524190, ranks sixth among all known primes by digit count and first among generalized Fermat primes. Searches predominantly target forms F_n(a) with large n (such as 20 or 21) and moderate a, or smaller n with very large a, as these balance computational tractability with the potential for record-breaking sizes. For instance, the second-largest known generalized Fermat prime is $4 \times 5^{11,786,358} + 1 (equivalent to F_1(2 \times 5^{5,893,179})), an 8,238,312-digit number found in 2024, which highlights the viability of low-n, high-a forms akin to Proth primes but strictly adhering to the power-of-two exponent. Other notable examples include primes like F_5(1162419), though these are significantly smaller and serve more as benchmarks for testing algorithms rather than records. Discovery relies on distributed computing networks, where volunteers contribute CPU/GPU cycles to sieve potential candidates using tools like sieving software (e.g., srieve or Genefer) to eliminate composites early, followed by primality proving via probabilistic tests such as the Lucas-Lehmer-Riesel (LLR) or primality proving (ECPP). For larger candidates, methods like Pollard's p-1 and the (ECM) are applied to factor out small divisors before full primality verification with OpenPFGW or , often requiring months of collective effort. PrimeGrid's GFNS, for example, has systematically searched n from 14 upward, completing sieving for n \leq 26 in ranges of k up to millions by mid-2025, with ongoing efforts targeting n up to 40 for smaller k intervals; over such primes have been found since the project's inception, though most are under 1 million digits. These discoveries illustrate the sustained interest in generalized Fermat primes within and , as they test the boundaries of primality proving for special-form numbers and contribute to databases like The Prime Pages, fostering improvements in algorithms that may apply to broader cryptographic or challenges. No larger primes of this form are anticipated in the near term without breakthroughs in quantum or specialized , maintaining the current record's prominence into late 2025.