Named after  Pierre de Fermat 

No. of known terms  5 
Conjectured no. of terms  5 
Subsequence of  Fermat numbers 
First terms  3, 5, 17, 257, 65537 
Largest known term  65537 
OEIS index  A019434 
In mathematics, a Fermat number, named after Pierre de Fermat, the first known to have studied them, is a positive integer of the form
where n is a nonnegative integer. The first few Fermat numbers are:
If 2^{k} + 1 is prime and k > 0, then k itself must be a power of 2, so 2^{k} + 1 is a Fermat number; such primes are called Fermat primes. As of 2023^{[update]}, the only known Fermat primes are F_{0} = 3, F_{1} = 5, F_{2} = 17, F_{3} = 257, and F_{4} = 65537 (sequence A019434 in the OEIS); heuristics suggest that there are no more.
The Fermat numbers satisfy the following recurrence relations:
for n ≥ 1,
for n ≥ 2. Each of these relations can be proved by mathematical induction. From the second equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor greater than 1. To see this, suppose that 0 ≤ i < j and F_{i} and F_{j} have a common factor a > 1. Then a divides both
and F_{j}; hence a divides their difference, 2. Since a > 1, this forces a = 2. This is a contradiction, because each Fermat number is clearly odd. As a corollary, we obtain another proof of the infinitude of the prime numbers: for each F_{n}, choose a prime factor p_{n}; then the sequence {p_{n}} is an infinite sequence of distinct primes.
Fermat numbers and Fermat primes were first studied by Pierre de Fermat, who conjectured that all Fermat numbers are prime. Indeed, the first five Fermat numbers F_{0}, ..., F_{4} are easily shown to be prime. Fermat's conjecture was refuted by Leonhard Euler in 1732 when he showed that
Euler proved that every factor of F_{n} must have the form k 2^{n+1} + 1 (later improved to k 2^{n+2} + 1 by Lucas) for n ≥ 2.
That 641 is a factor of F_{5} can be deduced from the equalities 641 = 2^{7} × 5 + 1 and 641 = 2^{4} + 5^{4}. It follows from the first equality that 2^{7} × 5 ≡ −1 (mod 641) and therefore (raising to the fourth power) that 2^{28} × 5^{4} ≡ 1 (mod 641). On the other hand, the second equality implies that 5^{4} ≡ −2^{4} (mod 641). These congruences imply that 2^{32} ≡ −1 (mod 641).
Fermat was probably aware of the form of the factors later proved by Euler, so it seems curious that he failed to follow through on the straightforward calculation to find the factor.^{[1]} One common explanation is that Fermat made a computational mistake.
There are no other known Fermat primes F_{n} with n > 4, but little is known about Fermat numbers for large n.^{[2]} In fact, each of the following is an open problem:
As of 2014^{[update]}, it is known that F_{n} is composite for 5 ≤ n ≤ 32, although of these, complete factorizations of F_{n} are known only for 0 ≤ n ≤ 11, and there are no known prime factors for n = 20 and n = 24.^{[4]} The largest Fermat number known to be composite is F_{18233954}, and its prime factor 7 × 2^{18233956} + 1 was discovered in October 2020.
Heuristics suggest that F_{4} is the last Fermat prime.
The prime number theorem implies that a random integer in a suitable interval around N is prime with probability 1 / ln N. If one uses the heuristic that a Fermat number is prime with the same probability as a random integer of its size, and that F_{5}, ..., F_{32} are composite, then the expected number of Fermat primes beyond F_{4} (or equivalently, beyond F_{32}) should be
One may interpret this number as an upper bound for the probability that a Fermat prime beyond F_{4} exists.
This argument is not a rigorous proof. For one thing, it assumes that Fermat numbers behave "randomly", but the factors of Fermat numbers have special properties. Boklan and Conway published a more precise analysis suggesting that the probability that there is another Fermat prime is less than one in a billion.^{[5]}
Anders Bjorn and Hans Riesel estimated the number of square factors of Fermat numbers from F_{5} onward as
in other words, there are unlikely to be any nonsquarefree Fermat numbers, and in general square factors of are very rare for large n.^{[6]}
Main article: Pépin's test 
Let be the nth Fermat number. Pépin's test states that for n > 0,
The expression can be evaluated modulo by repeated squaring. This makes the test a fast polynomialtime algorithm. But Fermat numbers grow so rapidly that only a handful of them can be tested in a reasonable amount of time and space.
There are some tests for numbers of the form k 2^{m} + 1, such as factors of Fermat numbers, for primality.
If N = F_{n} > 3, then the above Jacobi symbol is always equal to −1 for a = 3, and this special case of Proth's theorem is known as Pépin's test. Although Pépin's test and Proth's theorem have been implemented on computers to prove the compositeness of some Fermat numbers, neither test gives a specific nontrivial factor. In fact, no specific prime factors are known for n = 20 and 24.
Because of Fermat numbers' size, it is difficult to factorize or even to check primality. Pépin's test gives a necessary and sufficient condition for primality of Fermat numbers, and can be implemented by modern computers. The elliptic curve method is a fast method for finding small prime divisors of numbers. Distributed computing project Fermatsearch has found some factors of Fermat numbers. Yves Gallot's proth.exe has been used to find factors of large Fermat numbers. Édouard Lucas, improving Euler's abovementioned result, proved in 1878 that every factor of the Fermat number , with n at least 2, is of the form (see Proth number), where k is a positive integer. By itself, this makes it easy to prove the primality of the known Fermat primes.
Factorizations of the first twelve Fermat numbers are:
F_{0}  =  2^{1}  +  1  =  3 is prime  
F_{1}  =  2^{2}  +  1  =  5 is prime  
F_{2}  =  2^{4}  +  1  =  17 is prime  
F_{3}  =  2^{8}  +  1  =  257 is prime  
F_{4}  =  2^{16}  +  1  =  65,537 is the largest known Fermat prime  
F_{5}  =  2^{32}  +  1  =  4,294,967,297  
=  641 × 6,700,417 (fully factored 1732^{[7]})  
F_{6}  =  2^{64}  +  1  =  18,446,744,073,709,551,617 (20 digits)  
=  274,177 × 67,280,421,310,721 (14 digits) (fully factored 1855)  
F_{7}  =  2^{128}  +  1  =  340,282,366,920,938,463,463,374,607,431,768,211,457 (39 digits)  
=  59,649,589,127,497,217 (17 digits) × 5,704,689,200,685,129,054,721 (22 digits) (fully factored 1970)  
F_{8}  =  2^{256}  +  1  =  115,792,089,237,316,195,423,570,985,008,687,907,853,269,984,665,640,564,039,457,584,007,913,129, 639,937 (78 digits) 

=  1,238,926,361,552,897 (16 digits) × 93,461,639,715,357,977,769,163,558,199,606,896,584,051,237,541,638,188,580,280,321 (62 digits) (fully factored 1980)  
F_{9}  =  2^{512}  +  1  =  13,407,807,929,942,597,099,574,024,998,205,846,127,479,365,820,592,393,377,723,561,443,721,764,0 30,073,546,976,801,874,298,166,903,427,690,031,858,186,486,050,853,753,882,811,946,569,946,433,6 49,006,084,097 (155 digits) 

=  2,424,833 × 7,455,602,825,647,884,208,337,395,736,200,454,918,783,366,342,657 (49 digits) × 741,640,062,627,530,801,524,787,141,901,937,474,059,940,781,097,519,023,905,821,316,144,415,759, 504,705,008,092,818,711,693,940,737 (99 digits) (fully factored 1990)  
F_{10}  =  2^{1024}  +  1  =  179,769,313,486,231,590,772,930...304,835,356,329,624,224,137,217 (309 digits)  
=  45,592,577 × 6,487,031,809 × 4,659,775,785,220,018,543,264,560,743,076,778,192,897 (40 digits) × 130,439,874,405,488,189,727,484...806,217,820,753,127,014,424,577 (252 digits) (fully factored 1995)  
F_{11}  =  2^{2048}  +  1  =  32,317,006,071,311,007,300,714,8...193,555,853,611,059,596,230,657 (617 digits)  
=  319,489 × 974,849 × 167,988,556,341,760,475,137 (21 digits) × 3,560,841,906,445,833,920,513 (22 digits) × 173,462,447,179,147,555,430,258...491,382,441,723,306,598,834,177 (564 digits) (fully factored 1988) 
As of April 2023^{[update]}, only F_{0} to F_{11} have been completely factored.^{[4]} The distributed computing project Fermat Search is searching for new factors of Fermat numbers.^{[8]} The set of all Fermat factors is A050922 (or, sorted, A023394) in OEIS.
The following factors of Fermat numbers were known before 1950 (since then, digital computers have helped find more factors):
Year  Finder  Fermat number  Factor 

1732  Euler  
1732  Euler  (fully factored)  
1855  Clausen  
1855  Clausen  (fully factored)  
1877  Pervushin  
1878  Pervushin  
1886  Seelhoff  
1899  Cunningham  
1899  Cunningham  
1903  Western  
1903  Western  
1903  Western  
1903  Western  
1903  Cullen  
1906  Morehead  
1925  Kraitchik 
As of July 2023^{[update]}, 368 prime factors of Fermat numbers are known, and 324 Fermat numbers are known to be composite.^{[4]} Several new Fermat factors are found each year.^{[9]}
Like composite numbers of the form 2^{p} − 1, every composite Fermat number is a strong pseudoprime to base 2. This is because all strong pseudoprimes to base 2 are also Fermat pseudoprimes – i.e.,
for all Fermat numbers.
In 1904, Cipolla showed that the product of at least two distinct prime or composite Fermat numbers will be a Fermat pseudoprime to base 2 if and only if .^{[10]}
Lemma. — If n is a positive integer,
Theorem — If is an odd prime, then is a power of 2.
If is a positive integer but not a power of 2, it must have an odd prime factor , and we may write where .
By the preceding lemma, for positive integer ,
where means "evenly divides". Substituting , and and using that is odd,
and thus
Because , it follows that is not prime. Therefore, by contraposition must be a power of 2.
Theorem — A Fermat prime cannot be a Wieferich prime.
We show if is a Fermat prime (and hence by the above, m is a power of 2), then the congruence does not hold.
Since we may write . If the given congruence holds, then , and therefore
Hence , and therefore . This leads to , which is impossible since .
Theorem (Édouard Lucas) — Any prime divisor p of is of the form whenever n > 1.
Let G_{p} denote the group of nonzero integers modulo p under multiplication, which has order p − 1. Notice that 2 (strictly speaking, its image modulo p) has multiplicative order equal to in G_{p} (since is the square of which is −1 modulo F_{n}), so that, by Lagrange's theorem, p − 1 is divisible by and p has the form for some integer k, as Euler knew. Édouard Lucas went further. Since n > 1, the prime p above is congruent to 1 modulo 8. Hence (as was known to Carl Friedrich Gauss), 2 is a quadratic residue modulo p, that is, there is integer a such that Then the image of a has order in the group G_{p} and (using Lagrange's theorem again), p − 1 is divisible by and p has the form for some integer s.
In fact, it can be seen directly that 2 is a quadratic residue modulo p, since
Since an odd power of 2 is a quadratic residue modulo p, so is 2 itself.
A Fermat number cannot be a perfect number or part of a pair of amicable numbers. (Luca 2000)
The series of reciprocals of all prime divisors of Fermat numbers is convergent. (Křížek, Luca & Somer 2002)
If n^{n} + 1 is prime, there exists an integer m such that n = 2^{2m}. The equation n^{n} + 1 = F_{(2m+m)} holds in that case.^{[11]}^{[12]}
Let the largest prime factor of the Fermat number F_{n} be P(F_{n}). Then,
Main article: Constructible polygon 
Carl Friedrich Gauss developed the theory of Gaussian periods in his Disquisitiones Arithmeticae and formulated a sufficient condition for the constructibility of regular polygons. Gauss stated that this condition was also necessary,^{[13]} but never published a proof. Pierre Wantzel gave a full proof of necessity in 1837. The result is known as the Gauss–Wantzel theorem:
A positive integer n is of the above form if and only if its totient φ(n) is a power of 2.
Fermat primes are particularly useful in generating pseudorandom sequences of numbers in the range 1, ..., N, where N is a power of 2. The most common method used is to take any seed value between 1 and P − 1, where P is a Fermat prime. Now multiply this by a number A, which is greater than the square root of P and is a primitive root modulo P (i.e., it is not a quadratic residue). Then take the result modulo P. The result is the new value for the RNG.
This is useful in computer science, since most data structures have members with 2^{X} possible values. For example, a byte has 256 (2^{8}) possible values (0–255). Therefore, to fill a byte or bytes with random values, a random number generator that produces values 1–256 can be used, the byte taking the output value −1. Very large Fermat primes are of particular interest in data encryption for this reason. This method produces only pseudorandom values, as after P − 1 repetitions, the sequence repeats. A poorly chosen multiplier can result in the sequence repeating sooner than P − 1.
Numbers of the form with a, b any coprime integers, a > b > 0, are called generalized Fermat numbers. An odd prime p is a generalized Fermat number if and only if p is congruent to 1 (mod 4). (Here we consider only the case n > 0, so 3 = is not a counterexample.)
An example of a probable prime of this form is 1215^{131072} + 242^{131072} (found by Kellen Shenton).^{[14]}
By analogy with the ordinary Fermat numbers, it is common to write generalized Fermat numbers of the form as F_{n}(a). In this notation, for instance, the number 100,000,001 would be written as F_{3}(10). In the following we shall restrict ourselves to primes of this form, , such primes are called "Fermat primes base a". Of course, these primes exist only if a is even.
If we require n > 0, then Landau's fourth problem asks if there are infinitely many generalized Fermat primes F_{n}(a).
Because of the ease of proving their primality, generalized Fermat primes have become in recent years a topic for research within the field of number theory. Many of the largest known primes today are generalized Fermat primes.
Generalized Fermat numbers can be prime only for even a, because if a is odd then every generalized Fermat number will be divisible by 2. The smallest prime number with is , or 30^{32} + 1. Besides, we can define "half generalized Fermat numbers" for an odd base, a half generalized Fermat number to base a (for odd a) is , and it is also to be expected that there will be only finitely many half generalized Fermat primes for each odd base.
In this list, the generalized Fermat numbers () to an even a are , for odd a, they are . If a is a perfect power with an odd exponent (sequence A070265 in the OEIS), then all generalized Fermat number can be algebraic factored, so they cannot be prime.
See^{[15]}^{[16]} for even bases up to 1000, and^{[17]} for odd bases. For the smallest number such that is prime, see OEIS: A253242.
numbers such that is prime 
numbers such that is prime 
numbers such that is prime 
numbers such that is prime  

2  0, 1, 2, 3, 4, ...  18  0, ...  34  2, ...  50  ... 
3  0, 1, 2, 4, 5, 6, ...  19  1, ...  35  1, 2, 6, ...  51  1, 3, 6, ... 
4  0, 1, 2, 3, ...  20  1, 2, ...  36  0, 1, ...  52  0, ... 
5  0, 1, 2, ...  21  0, 2, 5, ...  37  0, ...  53  3, ... 
6  0, 1, 2, ...  22  0, ...  38  ...  54  1, 2, 5, ... 
7  2, ...  23  2, ...  39  1, 2, ...  55  ... 
8  (none)  24  1, 2, ...  40  0, 1, ...  56  1, 2, ... 
9  0, 1, 3, 4, 5, ...  25  0, 1, ...  41  4, ...  57  0, 2, ... 
10  0, 1, ...  26  1, ...  42  0, ...  58  0, ... 
11  1, 2, ...  27  (none)  43  3, ...  59  1, ... 
12  0, ...  28  0, 2, ...  44  4, ...  60  0, ... 
13  0, 2, 3, ...  29  1, 2, 4, ...  45  0, 1, ...  61  0, 1, 2, ... 
14  1, ...  30  0, 5, ...  46  0, 2, 9, ...  62  ... 
15  1, ...  31  ...  47  3, ...  63  ... 
16  0, 1, 2, ...  32  (none)  48  2, ...  64  (none) 
17  2, ...  33  0, 3, ...  49  1, ...  65  1, 2, 5, ... 
For the smallest even base a such that is prime, see OEIS: A056993.
bases a such that is prime (only consider even a)  OEIS sequence  

0  2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, ...  A006093 
1  2, 4, 6, 10, 14, 16, 20, 24, 26, 36, 40, 54, 56, 66, 74, 84, 90, 94, 110, 116, 120, 124, 126, 130, 134, 146, 150, 156, 160, 170, 176, 180, 184, ...  A005574 
2  2, 4, 6, 16, 20, 24, 28, 34, 46, 48, 54, 56, 74, 80, 82, 88, 90, 106, 118, 132, 140, 142, 154, 160, 164, 174, 180, 194, 198, 204, 210, 220, 228, ...  A000068 
3  2, 4, 118, 132, 140, 152, 208, 240, 242, 288, 290, 306, 378, 392, 426, 434, 442, 508, 510, 540, 542, 562, 596, 610, 664, 680, 682, 732, 782, ...  A006314 
4  2, 44, 74, 76, 94, 156, 158, 176, 188, 198, 248, 288, 306, 318, 330, 348, 370, 382, 396, 452, 456, 470, 474, 476, 478, 560, 568, 598, 642, ...  A006313 
5  30, 54, 96, 112, 114, 132, 156, 332, 342, 360, 376, 428, 430, 432, 448, 562, 588, 726, 738, 804, 850, 884, 1068, 1142, 1198, 1306, 1540, 1568, ...  A006315 
6  102, 162, 274, 300, 412, 562, 592, 728, 1084, 1094, 1108, 1120, 1200, 1558, 1566, 1630, 1804, 1876, 2094, 2162, 2164, 2238, 2336, 2388, ...  A006316 
7  120, 190, 234, 506, 532, 548, 960, 1738, 1786, 2884, 3000, 3420, 3476, 3658, 4258, 5788, 6080, 6562, 6750, 7692, 8296, 9108, 9356, 9582, ...  A056994 
8  278, 614, 892, 898, 1348, 1494, 1574, 1938, 2116, 2122, 2278, 2762, 3434, 4094, 4204, 4728, 5712, 5744, 6066, 6508, 6930, 7022, 7332, ...  A056995 
9  46, 1036, 1318, 1342, 2472, 2926, 3154, 3878, 4386, 4464, 4474, 4482, 4616, 4688, 5374, 5698, 5716, 5770, 6268, 6386, 6682, 7388, 7992, ...  A057465 
10  824, 1476, 1632, 2462, 2484, 2520, 3064, 3402, 3820, 4026, 6640, 7026, 7158, 9070, 12202, 12548, 12994, 13042, 15358, 17646, 17670, ...  A057002 
11  150, 2558, 4650, 4772, 11272, 13236, 15048, 23302, 26946, 29504, 31614, 33308, 35054, 36702, 37062, 39020, 39056, 43738, 44174, 45654, ...  A088361 
12  1534, 7316, 17582, 18224, 28234, 34954, 41336, 48824, 51558, 51914, 57394, 61686, 62060, 89762, 96632, 98242, 100540, 101578, 109696, ...  A088362 
13  30406, 71852, 85654, 111850, 126308, 134492, 144642, 147942, 150152, 165894, 176206, 180924, 201170, 212724, 222764, 225174, 241600, ...  A226528 
14  67234, 101830, 114024, 133858, 162192, 165306, 210714, 216968, 229310, 232798, 422666, 426690, 449732, 462470, 468144, 498904, 506664, ...  A226529 
15  70906, 167176, 204462, 249830, 321164, 330716, 332554, 429370, 499310, 524552, 553602, 743788, 825324, 831648, 855124, 999236, 1041870, ...  A226530 
16  48594, 108368, 141146, 189590, 255694, 291726, 292550, 357868, 440846, 544118, 549868, 671600, 843832, 857678, 1024390, 1057476, 1087540, ...  A251597 
17  62722, 130816, 228188, 386892, 572186, 689186, 909548, 1063730, 1176694, 1361244, 1372930, 1560730, 1660830, 1717162, 1722230, 1766192, ...  A253854 
18  24518, 40734, 145310, 361658, 525094, 676754, 773620, 1415198, 1488256, 1615588, 1828858, 2042774, 2514168, 2611294, 2676404, 3060772, ...  A244150 
19  75898, 341112, 356926, 475856, 1880370, 2061748, 2312092, 2733014, 2788032, 2877652, 2985036, 3214654, 3638450, 4896418, 5897794, ...  A243959 
20  919444, 1059094, 1951734, 1963736, ...  A321323 
The smallest bases b=b(n) such that b^{2n} + 1 (for given n= 0,1,2, ...) is prime are
Conversely, the smallest k=k(n) such that (2n)^{k} + 1 (for given n) is prime are
A more elaborate theory can be used to predict the number of bases for which will be prime for fixed . The number of generalized Fermat primes can be roughly expected to halve as is increased by 1.
It is also possible to construct generalized Fermat primes of the form . As in the case where b=1, numbers of this form will always be divisible by 2 if a+b is even, but it is still possible to define generalized halfFermat primes of this type. For the smallest prime of the form (for odd ), see also OEIS: A111635.
numbers such that is prime^{[18]}^{[6]}  

2  1  0, 1, 2, 3, 4, ... 
3  1  0, 1, 2, 4, 5, 6, ... 
3  2  0, 1, 2, ... 
4  1  0, 1, 2, 3, ... (equivalent to ) 
4  3  0, 2, 4, ... 
5  1  0, 1, 2, ... 
5  2  0, 1, 2, ... 
5  3  1, 2, 3, ... 
5  4  1, 2, ... 
6  1  0, 1, 2, ... 
6  5  0, 1, 3, 4, ... 
7  1  2, ... 
7  2  1, 2, ... 
7  3  0, 1, 8, ... 
7  4  0, 2, ... 
7  5  1, 4, 
7  6  0, 2, 4, ... 
8  1  (none) 
8  3  0, 1, 2, ... 
8  5  0, 1, 2, 
8  7  1, 4, ... 
9  1  0, 1, 3, 4, 5, ... (equivalent to ) 
9  2  0, 2, ... 
9  4  0, 1, ... (equivalent to ) 
9  5  0, 1, 2, ... 
9  7  2, ... 
9  8  0, 2, 5, ... 
10  1  0, 1, ... 
10  3  0, 1, 3, ... 
10  7  0, 1, 2, ... 
10  9  0, 1, 2, ... 
11  1  1, 2, ... 
11  2  0, 2, ... 
11  3  0, 3, ... 
11  4  1, 2, ... 
11  5  1, ... 
11  6  0, 1, 2, ... 
11  7  2, 4, 5, ... 
11  8  0, 6, ... 
11  9  1, 2, ... 
11  10  5, ... 
12  1  0, ... 
12  5  0, 4, ... 
12  7  0, 1, 3, ... 
12  11  0, ... 
The following is a list of the five largest known generalized Fermat primes.^{[19]} The whole top5 is discovered by participants in the PrimeGrid project.
Rank  Prime number  Generalized Fermat notation  Number of digits  Discovery date  ref. 

1  1963736^{1048576} + 1  F_{20}(1963736)  6,598,776  Sep 2022  ^{[20]} 
2  1951734^{1048576} + 1  F_{20}(1951734)  6,595,985  Aug 2022  ^{[21]} 
3  1059094^{1048576} + 1  F_{20}(1059094)  6,317,602  Nov 2018  ^{[22]} 
4  919444^{1048576} + 1  F_{20}(919444)  6,253,210  Sep 2017  ^{[23]} 
5  81 × 2^{20498148} + 1  F_{2}(3 × 2^{5124537})  6,170,560  Jun 2023  ^{[24]} 
On the Prime Pages one can find the current top 100 generalized Fermat primes.