Definition:Fermat Number

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Definition

A Fermat number is a natural number of the form $2^{\paren {2^n} } + 1$, where $n = 0, 1, 2, \ldots$.

The number $2^{\paren {2^n} } + 1$ is, in this context, often denoted $F_n$.


Sequence

The sequence of Fermat numbers begins:

\(\ds 2^{\paren {2^0} } + 1\) \(=\) \(\ds 3\)
\(\ds 2^{\paren {2^1} } + 1\) \(=\) \(\ds 5\)
\(\ds 2^{\paren {2^2} } + 1\) \(=\) \(\ds 17\)
\(\ds 2^{\paren {2^3} } + 1\) \(=\) \(\ds 257\)
\(\ds 2^{\paren {2^4} } + 1\) \(=\) \(\ds 65 \, 537\)
\(\ds 2^{\paren {2^5} } + 1\) \(=\) \(\ds 4 \, 294 \, 967 \, 297\)
\(\ds 2^{\paren {2^6} } + 1\) \(=\) \(\ds 18 \, 446 \, 744 \, 073 \, 709 \, 551 \, 617\)


Naming Conventions

The Fermat number $F_0$ is often referred to as the $1$st Fermat number, so (confusingly) this convention dictates that $F_n$ is the $n + 1$th Fermat number.

However, another convention is that $F_0$ can be referred to as the zeroth Fermat number, thus bringing the appellation in line such that $F_n$ is the $n$th Fermat number.

Both conventions are in place, sometimes in the same work.

For example, David Wells, in his Curious and Interesting Numbers, 2nd ed. of $1997$, refers to $5 = F_1$ in Section $5$ as the $2$nd Fermat number.

However, in Section $257$ he defines $F_3 = 2^{2^3} + 1 = 257$ as the $3$rd Fermat number.

Similarly, in Section $65,537$ he defines $F_4 = 2^{2^4} + 1 = 65 \, 537$ as the $4$th Fermat number, and so on.


Both of these naming conventions is more or less clumsy.

$\mathsf{Pr} \infty \mathsf{fWiki}$ takes the position that the cat has to jump one way or the other, and so uses the second of these conventions:

$F_n$ is the $n$th Fermat number.


Also see

  • Results about Fermat Numbers can be found here.


Source of Name

This entry was named for Pierre de Fermat.


Historical Note

In $1640$, Pierre de Fermat wrote to Bernard Frénicle de Bessy that $2^n + 1$ is composite if $n$ is divisible by an odd prime.

He also observed that the first $5$ numbers of the form $2^{2^n} + 1$ are all prime.

This led him to propose the Fermat Prime Conjecture: that all numbers of this form are prime.

On being unable to prove it, he sent the problem to Blaise Pascal, with the note:

I wouldn't ask you to work at it if I had been successful.

Pascal unfortunately did not take up the challenge.


The Fermat Prime Conjecture was proved false by Leonhard Paul Euler, who discovered the prime decomposition of the $6$th Fermat number $F_5$.

In $1877$, Ivan Mikheevich Pervushin proved that $F_{12}$ is divisible by $7 \times 2^{14} + 1 = 114 \, 689$, but was unable to completely factorise it.

In $1878$, he similarly found that $5 \times 2^{25} + 1$ is a divisor of $F_{23}$.

Fortuné Landry factorised $F_6$ in $1880$, in the process setting the still-unbroken record for finding the largest non-Mersenne prime number without the use of a computer.

In $1909$, James Caddall Morehead and Alfred E. Western reported in Bulletin of the American Mathematical Society that they had proved that $F_7$ and $F_8$ are not prime, but without having established what the prime factors are.

Prior to that, several divisors of various Fermat numbers had been identified, including $F_{73}$ by Morehead, who found the divisor $5 \times 2^{75} + 1$ in $1906$.

The prime factors of $F_7$ were finally discovered by Michael A. Morrison and John David Brillhart in $1970$:

$F_7 = \paren {116 \, 503 \, 103 \, 764 \, 643 \times 2^9 + 1} \paren {11 \, 141 \, 971 \, 095 \, 088 \, 142 \, 685 \times 2^9 + 1}$


One of the divisors of $F_8$ was found by Richard Peirce Brent and John Michael Pollard in $1981$:

$1 \, 238 \, 926 \, 361 \, 552 \, 897$


Some divisors of truly colossal Fermat numbers are known.

For example:

a divisor of $F_{1945}$ is known
$19 \times 2^{9450} + 1$ is a divisor of $F_{9448}$
$5 \times 2^{23 \, 473} + 1$ is a divisor of $F_{23 \, 471}$


Sources

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