Fermat Prime Conjecture

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False Conjecture

All numbers of the form $2^{\paren {2^n} } + 1$, where $n = 0, 1, 2, \ldots$ are prime.

This was postulated by Pierre de Fermat.


Refutation

Although true for $n = 0, 1, 2, 3, 4$, the conjecture fails for $n = 5$.

From Prime Decomposition of $5$th Fermat Number:

$2^{\paren {2^5} } + 1 = 641 \times 6 \, 700 \, 417$

$\blacksquare$


Also see


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 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 $F_{23 \, 471}$


Sources