# Definition:Fermat Prime

## Definition

A Fermat prime is a Fermat number, that is a number of the form $2^{\paren {2^n} } + 1$, which happens to be prime.

In fact, $2^{\paren {2^n} } + 1$ is prime for $n = 0, 1, 2, 3, 4$.

However, $2^{\paren {2^5} } + 1 = 2^{32} + 1$ is divisible by $641$, as was proved by Euler.

No Fermat prime for $n > 4$ has ever been discovered.

### Sequence

The sequence of Fermat primes begins:

 $\displaystyle 2^{\paren {2^0} } + 1$ $=$ $\displaystyle 3$ $\displaystyle 2^{\paren {2^1} } + 1$ $=$ $\displaystyle 5$ $\displaystyle 2^{\paren {2^2} } + 1$ $=$ $\displaystyle 17$ $\displaystyle 2^{\paren {2^3} } + 1$ $=$ $\displaystyle 257$ $\displaystyle 2^{\paren {2^4} } + 1$ $=$ $\displaystyle 65 \, 537$

No other Fermat primes are known.

## Also see

• Results about Fermat primes can be found here.

## Source of Name

This entry was named for Pierre de Fermat.

## Historical Note

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 C. 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 $F_{23 \, 471}$