Fermat number

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 non-negative integer. The first few Fermat numbers are: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457,....
If 2^k + 1 is prime and, then k itself must be a power of 2, so is a Fermat number; such primes are called Fermat primes., the only known Fermat primes are,,,, and .

Basic properties

The Fermat numbers satisfy the following recurrence relations:
for n ≥ 1,
for. Each of these relations can be proved by mathematical induction. From the second equation, we can deduce Goldbach's theorem : no two Fermat numbers share a common integer factor greater than 1. To see this, suppose that and F_i and F_j have a common factor. Then a divides both
and F_j; hence a divides their difference, 2. Since, this forces. This is a contradiction, because each Fermat number is clearly odd. As a corollary, we obtain another proof of the prime numbers">Prime number">prime numbers: for each F_n, choose a prime factor p_n; then the sequence is an infinite sequence of distinct primes.

Further properties

No Fermat prime can be expressed as the difference of two pth powers, where p is an odd prime.
With the exception of F₀ and F₁, the last decimal digit of a Fermat number is 7.
The sum of the reciprocals of all the Fermat numbers is irrational.
Primality

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 F₀,..., F₄ are easily shown to be prime. Fermat's conjecture was refuted by Leonhard Euler in 1732 when he showed, by dividing by 641 that
Euler proved that every factor of F_n must have the form for.
That 641 is a factor of F₅ can be deduced, in hindsight, as follows: From the equalities 641 = 2⁷ × 5 + 1 and 641 = 2⁴ + 5⁴. It follows from the first equality that 2⁷ × 5 ≡ −1 and therefore that 2²⁸ × 5⁴ ≡ 1 . On the other hand, the second equality implies that 5⁴ ≡ −2⁴ . These congruences imply that 2³² ≡ −1 .
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. One common explanation is that Fermat made a computational mistake.
There are no other known Fermat primes F_n with, but little is known about Fermat numbers for large n. In fact, each of the following is an open problem:

Is F_n composite for all ?
Are there infinitely many Fermat primes?
Are there infinitely many composite Fermat numbers?
Does a Fermat number exist that is not square-free?

, it is known that F_n is composite for, although of these, complete factorizations of F_n are known only for, and there are no known prime factors for and. The largest Fermat number known to be composite is F_18233954, and its prime factor was discovered in October 2020.

Heuristic arguments

Heuristics suggest that F₄ 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 F₅,..., F₃₂ are composite, then the expected number of Fermat primes beyond F₄ should be
One may interpret this number as an upper bound for the probability that a Fermat prime beyond F₄ 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.
Anders Bjorn and Hans Riesel estimated the number of square factors of Fermat numbers from F₅ onward as
in other words, there are unlikely to be any non-squarefree Fermat numbers, and in general square factors of are very rare for large n.

Equivalent conditions

Let be the nth Fermat number. Pépin's test states that for,
The expression can be evaluated modulo by repeated squaring. This makes the test a fast polynomial-time 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, such as factors of Fermat numbers, for primality.
If, then the above Jacobi symbol is always equal to −1 for, 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 and 24.

Factorization

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 has been used to find factors of large Fermat numbers. Édouard Lucas, improving Euler's above-mentioned result, proved in 1878 that every factor of the Fermat number, with n at least 2, is of the form , 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 12 Fermat numbers are:
, only F₀ to F₁₁ have been completely factored. The distributed computing project Fermat Search is searching for new factors of Fermat numbers. The set of all Fermat factors is A050922 in OEIS.
The following factors of Fermat numbers were known before 1950 :

Year	Finder	Fermat number	Factor
1732	Euler
1732	Euler
1855	Clausen (mathematician)\|Clausen]
1855	Clausen
1877	Pervushin
1878	Pervushin
1886	Seelhoff
1899	Cunningham
1899	Cunningham
1903	Western
1903	Western
1903	Western
1903	Western
1903	Cullen (mathematician)\|Cullen]
1906	Morehead
1925	Kraitchik

, 375 prime factors of Fermat numbers are known, and 330 Fermat numbers are known to be composite. Several new Fermat factors are found each year.

Pseudoprimes and Fermat numbers

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.

Other theorems about Fermat numbers

A Fermat number cannot be a perfect number or part of a pair of amicable numbers.
The series of reciprocals of all prime divisors of Fermat numbers is convergent.
If is prime and, there exists an integer m such that. The equation
holds in that case.
Let the largest prime factor of the Fermat number F_n be P. Then,

Relationship to constructible polygons

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, 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 function|totient] φ is a power of 2.

Applications of Fermat numbers

Pseudorandom number generation

Fermat primes are particularly useful in generating pseudo-random 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, 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. 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 possible values. 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 repetitions, the sequence repeats. A poorly chosen multiplier can result in the sequence repeating sooner than.

Generalized Fermat numbers

Numbers of the form with a, b any coprime integers,, 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).
An example of a probable prime of this form is 200²⁶²¹⁴⁴ + 119²⁶²¹⁴⁴.
By analogy with the ordinary Fermat numbers, it is common to write generalized Fermat numbers of the form as F_n. In this notation, for instance, the number 100,000,001 would be written as F₃. 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, then Landau's fourth problem asks if there are infinitely many generalized Fermat primes F_n.

Generalized Fermat primes of the form 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, because if is odd then every generalized Fermat number will be divisible by 2. The smallest prime number with is, or 30³² + 1. Besides, we can define "half generalized Fermat numbers" for an odd base, a half generalized Fermat number to base 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 are, for odd, they are. If is a perfect power with an odd exponent, then all generalized Fermat number can be algebraic factored, so they cannot be prime.
See for even bases up to 1000, and for odd bases. For the smallest number such that is prime, see.

	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		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		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		48	2,...	64
17	2,...	33	0, 3,...	49	1,...	65	1, 2, 5,...

For the smallest even base such that is prime, see.
The generalized Fermat prime F₁₄ is the largest known generalized Fermat prime in bases b ≤ 1000, it is proven prime by elliptic curve primality proving.

	bases such that is prime	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,...
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,...
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,...
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,...
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,...
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,...
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,...
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,...
8	278, 614, 892, 898, 1348, 1494, 1574, 1938, 2116, 2122, 2278, 2762, 3434, 4094, 4204, 4728, 5712, 5744, 6066, 6508, 6930, 7022, 7332,...
9	46, 1036, 1318, 1342, 2472, 2926, 3154, 3878, 4386, 4464, 4474, 4482, 4616, 4688, 5374, 5698, 5716, 5770, 6268, 6386, 6682, 7388, 7992,...
10	824, 1476, 1632, 2462, 2484, 2520, 3064, 3402, 3820, 4026, 6640, 7026, 7158, 9070, 12202, 12548, 12994, 13042, 15358, 17646, 17670,...
11	150, 2558, 4650, 4772, 11272, 13236, 15048, 23302, 26946, 29504, 31614, 33308, 35054, 36702, 37062, 39020, 39056, 43738, 44174, 45654,...
12	1534, 7316, 17582, 18224, 28234, 34954, 41336, 48824, 51558, 51914, 57394, 61686, 62060, 89762, 96632, 98242, 100540, 101578, 109696,...
13	30406, 71852, 85654, 111850, 126308, 134492, 144642, 147942, 150152, 165894, 176206, 180924, 201170, 212724, 222764, 225174, 241600,...
14	67234, 101830, 114024, 133858, 162192, 165306, 210714, 216968, 229310, 232798, 422666, 426690, 449732, 462470, 468144, 498904, 506664,...
15	70906, 167176, 204462, 249830, 321164, 330716, 332554, 429370, 499310, 524552, 553602, 743788, 825324, 831648, 855124, 999236, 1041870, 1074542, 1096382, 1113768, 1161054, 1167528, 1169486, 1171824, 1210354, 1217284, 1277444, 1519380, 1755378, 1909372, 1922592, 1986700,...
16	48594, 108368, 141146, 189590, 255694, 291726, 292550, 357868, 440846, 544118, 549868, 671600, 843832, 857678, 1024390, 1057476, 1087540, 1266062, 1361846, 1374038, 1478036, 1483076, 1540550, 1828502, 1874512, 1927034, 1966374,...
17	62722, 130816, 228188, 386892, 572186, 689186, 909548, 1063730, 1176694, 1361244, 1372930, 1560730, 1660830, 1717162, 1722230, 1766192, 1955556, 2194180, 2280466, 2639850, 3450080, 3615210, 3814944, 4085818, 4329134, 4893072, 4974408,...
18	24518, 40734, 145310, 361658, 525094, 676754, 773620, 1415198, 1488256, 1615588, 1828858, 2042774, 2514168, 2611294, 2676404, 3060772, 3547726, 3596074, 3673932, 3853792, 3933508, 4246258, 4489246,...
19	75898, 341112, 356926, 475856, 1880370, 2061748, 2312092, 2733014, 2788032, 2877652, 2985036, 3214654, 3638450, 4896418, 5897794, 6339004, 8630170, 9332124, 10913140, 11937916, 12693488, 12900356,...
20	919444, 1059094, 1951734, 1963736, 3843236, 5336284,...
21	2524190,...

The smallest even base b such that F_n = b^2ⁿ + 1 is prime are
The smallest odd base b such that F_n = /2 is prime are
Conversely, the smallest k such that ^k + 1 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.

Generalized Fermat primes of the form F_n(''a'', ''b'')

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 half-Fermat primes of this type. For the smallest prime of the form , see also.

		numbers such that is prime
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,...
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
8	3	0, 1, 2,...
8	5	0, 1, 2,
8	7	1, 4,...
9	1	0, 1, 3, 4, 5,...
9	2	0, 2,...
9	4	0, 1,...
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,...

Largest known generalized Fermat primes

The following is a list of the ten largest known generalized Fermat primes. The whole top-10 is discovered by participants in the PrimeGrid project.

Rank	Prime number	Generalized Fermat notation	Number of digits	Discovery date	ref.
1	2524190^2097152 + 1	F₂₁	13,426,224	Oct 2025
2	4×5^11786358 + 1	F₁	8,238,312	Oct 2024
3	5336284^1048576 + 1	F₂₀	7,054,022	Nov 2025
4	3843236^1048576 + 1	F₂₀	6,904,556	Dec 2024
5	1963736^1048576 + 1	F₂₀	6,598,776	Sep 2022
6	1951734^1048576 + 1	F₂₀	6,595,985	Aug 2022
7	1059094^1048576 + 1	F₂₀	6,317,602	Nov 2018
8	919444^1048576 + 1	F₂₀	6,253,210	Sep 2017
9	81×2^20498148 + 1	F₂	6,170,560	Jun 2023
10	4×5^8431178 + 1	F₁	5,893,142	Jan 2024

On the Prime Pages one can find the and the .