Fermat number#Primality

The Fermat numbers satisfy the following recurrence relations:

:$$

F_{n} = (F_{n-1}-1)^{2}+1

:$$

F_{n} = F_{0} \cdots F_{n-1} + 2

for n ≥ 1,

:$$

F_{n} = F_{n-1} + 2^{2^{n-1}}F_{0} \cdots F_{n-2}

:$$

F_{n} = F_{n-1}^2 - 2(F_{n-2}-1)^2

for {{nowrap|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 {{nowrap|0 ≤ i < j}} and F_i and F_j have a common factor {{nowrap|a > 1}}. Then a divides both

:$F\_\{0\}\; \backslash cdots\; F\_\{j-1\}$

and F_j; hence a divides their difference, 2. Since {{nowrap|a > 1}}, this forces {{nowrap|1=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 {{mset|p_n}} is an infinite sequence of distinct primes.

• 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 {{OEIS|id=A051158}} is irrational. (Solomon W. Golomb, 1963)

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 that

:$F\_\{5\}\; =\; 2^\{2^5\}\; +\; 1\; =\; 2^\{32\}\; +\; 1\; =\; 4294967297\; =\; 641\; \backslash times\; 6700417.$

Euler proved that every factor of F_n must have the form {{nowrap|k{{space|hair}}2ⁿ⁺¹ + 1}} (later improved to {{nowrap|k{{space|hair}}2ⁿ⁺² + 1}} by Lucas) for {{nowrap|n ≥ 2}}.

That 641 is a factor of F₅ can be deduced from the equalities 641 = 2⁷ × 5 + 1 and 641 = 2⁴ + 5⁴. It follows from the first equality that 2⁷ × 5 ≡ −1 (mod 641) and therefore (raising to the fourth power) that 2²⁸ × 5⁴ ≡ 1 (mod 641). On the other hand, the second equality implies that 5⁴ ≡ −2⁴ (mod 641). These congruences imply that 2³² ≡ −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.{{Harvnb|Křížek|Luca|Somer|2001|p=38, Remark 4.15}} One common explanation is that Fermat made a computational mistake.

There are no other known Fermat primes F_n with {{nowrap|n > 4}}, but little is known about Fermat numbers for large n.Chris Caldwell, [http://primes.utm.edu/links/theory/special_forms/ "Prime Links++: special forms"] {{webarchive|url=https://web.archive.org/web/20131224224552/http://primes.utm.edu/links/theory/special_forms/ |date=2013-12-24 }} at The Prime Pages. In fact, each of the following is an open problem:

• Is F_n composite for all {{nowrap|n > 4}}?
• Are there infinitely many Fermat primes? (Eisenstein 1844{{Harvnb|Ribenboim|1996|p=88}}.)
• Are there infinitely many composite Fermat numbers?
• Does a Fermat number exist that is not square-free?

{{As of|2024}}, it is known that F_n is composite for {{nowrap|5 ≤ n ≤ 32}}, although of these, complete factorizations of F_n are known only for {{nowrap|0 ≤ n ≤ 11}}, and there are no known prime factors for {{nowrap|1=n = 20}} and {{nowrap|1=n = 24}}. The largest Fermat number known to be composite is F_18233954, and its prime factor {{nowrap|7 × 2^18233956 + 1}} was discovered in October 2020.

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{{space|hair}}/{{space|hair}}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 (or equivalently, beyond F₃₂) 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.{{Cite journal |last1=Boklan |first1=Kent D. |last2=Conway |first2=John H. |date=2017 |title=Expect at most one billionth of a new Fermat Prime! |journal=The Mathematical Intelligencer |volume=39 |issue=1 |pages=3–5 |arxiv=1605.01371 |doi=10.1007/s00283-016-9644-3|s2cid=119165671 }}

Anders Bjorn and Hans Riesel estimated the number of square factors of Fermat numbers from F₅ onward as

:$$

\sum_{n \ge 5} \sum_{k \ge 1} \frac{1}{k (k 2^n + 1) \ln(k 2^n)} < \frac{\pi^2}{6 \ln 2} \sum_{n \ge 5} \frac{1}{n 2^n} \approx 0.02576;

in other words, there are unlikely to be any non-squarefree Fermat numbers, and in general square factors of $a^\{2^n\}\; +\; b^\{2^n\}$ are very rare for large n.{{cite journal |last1=Björn |first1=Anders |last2=Riesel |first2=Hans |title=Factors of generalized Fermat numbers |journal=Mathematics of Computation |date=1998 |volume=67 |issue=221 |pages=441–446 |doi=10.1090/S0025-5718-98-00891-6 |url=https://www.ams.org/journals/mcom/1998-67-221/S0025-5718-98-00891-6/ |language=en |issn=0025-5718|doi-access=free }}

{{Main|Pépin's test}}

Let $F\_n=2^\{2^n\}+1$ be the nth Fermat number. Pépin's test states that for {{nowrap|n > 0}},

:$F\_n$ is prime if and only if $3^\{(F\_n-1)/2\}\backslash equiv-1\backslash pmod\{F\_n\}.$

The expression $3^\{(F\_n-1)/2\}$ can be evaluated modulo $F\_n$ 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 {{nowrap|k{{space|hair}}2^m + 1}}, such as factors of Fermat numbers, for primality.

:Proth's theorem (1878). Let {{nowrap|1=N = k{{space|hair}}2^m + 1}} with odd {{nowrap|k < 2^m}}. If there is an integer a such that

:: $a^\{(N-1)/2\}\; \backslash equiv\; -1\backslash pmod\{N\}$

:then $N$ is prime. Conversely, if the above congruence does not hold, and in addition

:: $\backslash left(\backslash frac\{a\}\{N\}\backslash right)=-1$ (See Jacobi symbol)

:then $N$ is composite.

If {{nowrap|1=N = F_n > 3}}, then the above Jacobi symbol is always equal to −1 for {{nowrap|1=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 {{nowrap|1=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 above-mentioned result, proved in 1878 that every factor of the Fermat number $F\_n$, with n at least 2, is of the form $k\backslash times2^\{n+2\}+1$ (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 12 Fermat numbers are:

:

{{As of|2023|4}}, only F₀ to F₁₁ have been completely factored. The distributed computing project Fermat Search is searching for new factors of Fermat numbers.{{cite web|url=http://www.fermatsearch.org/|title=:: F E R M A T S E A R C H . O R G :: Home page|website=www.fermatsearch.org|access-date=7 April 2018}} 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):

{{As of|2024|12}}, 371 prime factors of Fermat numbers are known, and 324 Fermat numbers are known to be composite.{{Citation |first=Wilfrid |last=Keller |url=http://www.prothsearch.com/fermat.html#Summary |title=Prime Factors of Fermat Numbers |work=ProthSearch.com |date=January 18, 2021|access-date=January 19, 2021}} Several new Fermat factors are found each year.{{cite web |url=http://www.fermatsearch.org/news.html |title=::FERMATSEARCH.ORG:: News |website=www.fermatsearch.org |access-date=7 April 2018 }}

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.,

:$2^\{F\_n-1\}\; \backslash equiv\; 1\; \backslash pmod\{F\_n\}$

for all Fermat numbers.{{Cite book |last=Schroeder |first=M. R. |url=https://www.worldcat.org/title/ocm61430240 |title=Number theory in science and communication: with applications in cryptography, physics, digital information, computing, and self-similarity |date=2006 |publisher=Springer |isbn=978-3-540-26596-2 |edition=4th |series=Springer series in information sciences |location=Berlin ; New York |pages=216 |oclc=ocm61430240}}

In 1904, Cipolla showed that the product of at least two distinct prime or composite Fermat numbers $F\_\{a\}\; F\_\{b\}\; \backslash dots\; F\_\{s\},$ $a\; >\; b\; >\; \backslash dots\; >\; s\; >\; 1$ will be a Fermat pseudoprime to base 2 if and only if $2^s\; >\; a$.{{cite book|url=https://books.google.com/books?id=hgfSBwAAQBAJ&q=cipolla+fermat+1904&pg=PA132|title=17 Lectures on Fermat Numbers: From Number Theory to Geometry|first1=Michal|last1=Krizek|first2=Florian|last2=Luca|first3=Lawrence|last3=Somer|date=14 March 2013|publisher=Springer Science & Business Media|access-date=7 April 2018|via=Google Books|isbn=9780387218502}}

{{math theorem|name=Lemma.|If n is a positive integer,

::$a^n-b^n=(a-b)\backslash sum\_\{k=0\}^\{n-1\}\; a^kb^\{n-1-k\}.$

{{math proof|

$\backslash begin\{align\}$

(a-b)\sum_{k=0}^{n-1}a^kb^{n-1-k} &=\sum_{k=0}^{n-1}a^{k+1}b^{n-1-k}-\sum_{k=0}^{n-1}a^kb^{n-k}\\

&=a^n+\sum_{k=1}^{n-1}a^kb^{n-k}-\sum_{k=1}^{n-1}a^kb^{n-k}-b^n\\

&=a^n-b^n

\end{align}

}}

}}

{{math theorem| If $2^k+1$ is an odd prime, then $k$ is a power of 2.

{{math proof|If $k$ is a positive integer but not a power of 2, it must have an odd prime factor $s\; >\; 2$, and we may write $k=\; rs$ where .

By the preceding lemma, for positive integer $m$,

:$(a-b)\; \backslash mid\; (a^m-b^m)$

where $\backslash mid$ means "evenly divides". Substituting $a\; =\; 2^r,\; b\; =\; -1$, and $m\; =\; s$ and using that $s$ is odd,

:$(2^r+1)\; \backslash mid\; (2^\{rs\}+1),$

and thus

:$(2^r+1)\; \backslash mid\; (2^k+1).$

Because , it follows that $2^k+1$ is not prime. Therefore, by contraposition $k$ must be a power of 2.

}}}}

{{math theorem| A Fermat prime cannot be a Wieferich prime.

{{math proof| We show if $p=2^m+1$ is a Fermat prime (and hence by the above, m is a power of 2), then the congruence $2^\{p-1\}\; \backslash equiv\; 1\; \backslash bmod\; \{p^2\}$ does not hold.

Since $2m\; |p-1$ we may write $p-1=2m\backslash lambda$. If the given congruence holds, then $p^2|2^\{2m\backslash lambda\}-1$, and therefore

:$0\; \backslash equiv\; \backslash frac\{2^\{2m\backslash lambda\}-1\}\{2^m+1\}=(2^m-1)\backslash left(1+2^\{2m\}+2^\{4m\}+\backslash cdots\; +2^\{2(\backslash lambda-1)m\}\; \backslash right)\; \backslash equiv\; -2\backslash lambda\; \backslash pmod\; \{2^m+1\}.$

Hence $2^m+1|2\backslash lambda$, and therefore $2\backslash lambda\; \backslash geq\; 2^m+1$. This leads to $p-1\; \backslash geq\; m(2^m+1)$, which is impossible since $m\; \backslash geq\; 2$.

}}}}

{{math theorem|note=Édouard Lucas| Any prime divisor p of $F\_n\; =\; 2^\{2^n\}+1$ is of the form $k2^\{n+2\}+1$ whenever {{nowrap|n > 1}}.

{{math proof|title=Sketch of proof|Let G_p denote the group of non-zero integers modulo p under multiplication, which has order {{nowrap|p − 1}}. Notice that 2 (strictly speaking, its image modulo p) has multiplicative order equal to $2^\{n+1\}$ in G_p (since $2^\{2^\{n+1\}\}$ is the square of $2^\{2^n\}$ which is −1 modulo F_n), so that, by Lagrange's theorem, {{nowrap|p − 1}} is divisible by $2^\{n+1\}$ and p has the form $k2^\{n+1\}\; +\; 1$ for some integer k, as Euler knew. Édouard Lucas went further. Since {{nowrap|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 $p|a^2-2.$ Then the image of a has order $2^\{n+2\}$ in the group G_p and (using Lagrange's theorem again), {{nowrap|p − 1}} is divisible by $2^\{n+2\}$ and p has the form $s2^\{n+2\}\; +\; 1$ for some integer s.

In fact, it can be seen directly that 2 is a quadratic residue modulo p, since

:$\backslash left(1\; +2^\{2^\{n-1\}\}\; \backslash right)^\{2\}\; \backslash equiv\; 2^\{1+2^\{n-1\}\}\; \backslash pmod\; p.$

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. {{harv|Luca|2000}}

The series of reciprocals of all prime divisors of Fermat numbers is convergent. {{harv|Křížek|Luca|Somer|2002}}

If {{nowrap|nⁿ + 1}} is prime and $n\; \backslash ge\; 2$, there exists an integer m such that {{nowrap|1=n = 2^2m}}. The equation

{{nowrap|1=nⁿ + 1 = F_(2^m+m)}}

holds in that case.Jeppe Stig Nielsen, [http://jeppesn.dk/nton.html "S(n) = n^n + 1"].{{MathWorld|urlname=SierpinskiNumberoftheFirstKind|title=Sierpiński Number of the First Kind}}

Let the largest prime factor of the Fermat number F_n be P(F_n). Then,

:$P(F\_n)\; \backslash ge\; 2^\{n+2\}(4n+9)\; +\; 1.$ {{harv|Grytczuk|Luca|Wójtowicz|2001}}

File:Constructible polygon set.svg

{{Main|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,{{cite book |last1=Gauss |first1=Carl Friedrich |title=Disquisitiones arithmeticae |date=1966 |publisher=Yale University Press |location=New Haven and London |pages=458–460 |url=https://archive.org/details/disquisitionesar0000carl/ |access-date=25 January 2023}} but never published a proof. Pierre Wantzel gave a full proof of necessity in 1837. The result is known as the Gauss–Wantzel theorem:

: An n-sided regular polygon can be constructed with compass and straightedge if and only if n is either a power of 2 or the product of a power of 2 and distinct Fermat primes: in other words, if and only if n is of the form {{nowrap|1=n = 2^k}} or {{nowrap|1=n = 2^kp₁p₂...p_s}}, where k, s are nonnegative integers and the p_i are distinct Fermat primes.

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 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 {{nowrap|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.

: $V\_\{j+1\}\; =\; (A\; \backslash times\; V\_j)\; \backslash bmod\; P$ (see linear congruential generator)

This is useful in computer science, since most data structures have members with 2^X possible values. For example, a byte has 256 (2⁸) 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 {{nowrap|P − 1}} repetitions, the sequence repeats. A poorly chosen multiplier can result in the sequence repeating sooner than {{nowrap|P − 1}}.

Numbers of the form $\backslash frac\{a^\{2^n\}+b^\{2^n\}\}\{gcd(a+b,2)\}$ with a, b any coprime integers, {{nowrap|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 {{nowrap|n > 0}}, so {{nowrap|1=3 = $2^\{2^\{0\}\}\; \backslash !+\; 1$}} is not a counterexample.)

An example of a probable prime of this form is 200²⁶²¹⁴⁴ + 119²⁶²¹⁴⁴ (found by Kellen Shenton).[http://www.primenumbers.net/prptop/searchform.php?form=x%5E262144%2By%5E262144&action=Search PRP Top Records, search for x^262144+y^262144], by Henri & Renaud Lifchitz.

By analogy with the ordinary Fermat numbers, it is common to write generalized Fermat numbers of the form $a^\{2^\{\; \backslash overset\{n\}\; \{\}\}\}\; \backslash !\backslash !+\; 1$ as F_n(a). In this notation, for instance, the number 100,000,001 would be written as F₃(10). In the following we shall restrict ourselves to primes of this form, $a^\{2^\{\; \backslash overset\{n\}\; \{\}\}\}\; \backslash !\backslash !+\; 1$, such primes are called "Fermat primes base a". Of course, these primes exist only if a is even.

If we require {{nowrap|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 {{mvar|a}}, because if {{mvar|a}} is odd then every generalized Fermat number will be divisible by 2. The smallest prime number $F\_n(a)$ with $n>4$ is $F\_5(30)$, or 30³² + 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 $\backslash frac\{a^\{2^n\}\; \backslash !+\; 1\}\{2\}$, 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 ($F\_n(a)$) to an even {{mvar|a}} are $a^\{2^n\}\; \backslash !+\; 1$, for odd {{mvar|a}}, they are $\backslash frac\{a^\{2^n\}\; \backslash !\backslash !+\; 1\}\{2\}$. If {{mvar|a}} is a perfect power with an odd exponent {{OEIS|id=A070265}}, then all generalized Fermat number can be algebraic factored, so they cannot be prime.

See{{cite web|url=http://jeppesn.dk/generalized-fermat.html|title=Generalized Fermat Primes|website=jeppesn.dk|access-date=7 April 2018}}{{cite web|url=http://www.noprimeleftbehind.net/crus/GFN-primes.htm|title=Generalized Fermat primes for bases up to 1030|website=noprimeleftbehind.net|access-date=7 April 2018}} for even bases up to 1000, and{{cite web|url=http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt|title=Generalized Fermat primes in odd bases|website=fermatquotient.com|access-date=7 April 2018}} for odd bases. For the smallest number $n$ such that $F\_n(a)$ is prime, see {{oeis|id=A253242}}.

For the smallest even base {{mvar|a}} such that $F\_n(a)$ is prime, see {{oeis|id=A056993}}.

The generalized Fermat prime F₁₄(71) is the largest known generalized Fermat prime in bases b ≤ 1000, it is proven prime by elliptic curve primality proving.[https://factordb.com/index.php?id=1100000000213085670 The entry of the generalized Fermat prime F₁₄(71) in the online factor database]

The smallest even base b such that F_n(b) = b²ⁿ + 1 (for given n = 0, 1, 2, ...) is prime are

:2, 2, 2, 2, 2, 30, 102, 120, 278, 46, 824, 150, 1534, 30406, 67234, 70906, 48594, 62722, 24518, 75898, 919444, ... {{OEIS|id=A056993}}

The smallest odd base b such that F_n(b) = (b²ⁿ + 1)/2 (for given n = 0, 1, 2, ...) is prime (or probable prime) are

:3, 3, 3, 9, 3, 3, 3, 113, 331, 513, 827, 799, 3291, 5041, 71, 220221, 23891, 11559, 187503, 35963, ... {{OEIS|id=A275530}}

Conversely, the smallest k such that (2n)^k + 1 (for given n) is prime are

:1, 1, 1, 0, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 0, 4, 1, ... (The next term is unknown) {{OEIS|id=A079706}} (also see {{oeis|id=A228101}} and {{oeis|id=A084712}})

A more elaborate theory can be used to predict the number of bases for which $F\_n(a)$ will be prime for fixed $n$. The number of generalized Fermat primes can be roughly expected to halve as $n$ is increased by 1.

It is also possible to construct generalized Fermat primes of the form $a^\{2^n\}\; +\; b^\{2^n\}$. 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 $F\_n(a,b)$ (for odd $a+b$), see also {{oeis|id=A111635}}.

The following is a list of the ten largest known generalized Fermat primes.{{cite web|title=The Top Twenty: Generalized Fermat|url=http://primes.utm.edu/top20/page.php?id=12|work=The Prime Pages|first=Chris K.|last=Caldwell|access-date=5 October 2024}} The whole top-10 is discovered by participants in the PrimeGrid project.

On the Prime Pages one can find the [https://t5k.org/top20/page.php?id=12 current top 20 generalized Fermat primes] and the [https://t5k.org/primes/search.php?Comment=Generalized+Fermat&OnList=all&Number=100&Style=HTML current top 100 generalized Fermat primes].

• Constructible polygon: which regular polygons are constructible partially depends on Fermat primes.
• Double exponential function
• Lucas' theorem
• Mersenne prime
• Pierpont prime
• Primality test
• Proth's theorem
• Pseudoprime
• Sierpiński number
• Sylvester's sequence

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• {{citation |last=Yabuta |first=M. |journal=Fibonacci Quarterly |pages=439–443 |title=A simple proof of Carmichael's theorem on primitive divisors |url=http://www.fq.math.ca/Scanned/39-5/yabuta.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.fq.math.ca/Scanned/39-5/yabuta.pdf |archive-date=2022-10-09 |url-status=live |volume=39 |year=2001 |issue=5 |doi=10.1080/00150517.2001.12428701 }}

• Chris Caldwell, [http://primes.utm.edu/glossary/page.php?sort=FermatNumber The Prime Glossary: Fermat number] at The Prime Pages.
• Luigi Morelli, [http://www.fermatsearch.org/history.html History of Fermat Numbers]
• John Cosgrave, [http://johnbcosgrave.com/archive/fermat6.htm Unification of Mersenne and Fermat Numbers]
• Wilfrid Keller, [http://www.prothsearch.com/fermat.html Prime Factors of Fermat Numbers]
• {{MathWorld|title=Fermat Number|urlname=FermatNumber}}
• {{MathWorld|title=Fermat Prime|urlname=FermatPrime}}
• {{MathWorld|title=Generalized Fermat Number|urlname=GeneralizedFermatNumber}}
• Yves Gallot, [http://pagesperso-orange.fr/yves.gallot/primes/index.html Generalized Fermat Prime Search]
• Mark S. Manasse, [https://groups.google.com/forum/#!topic/sci.math/7usZOcN2_zc Complete factorization of the ninth Fermat number] (original announcement)
• Peyton Hayslette, [http://www.primegrid.com/download/GFN-341112_524288.pdf Largest Known Generalized Fermat Prime Announcement]

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Category:Constructible polygons

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