# Fermat Prime Conjecture

## 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

- Prime Decomposition of $5$th Fermat Number
- Prime Decomposition of $6$th Fermat Number
- Prime Decomposition of $7$th Fermat Number
- Prime Decomposition of $8$th Fermat Number
- Prime Decomposition of $9$th Fermat Number

## 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

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{IV}$: The Prince of Amateurs - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $257$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**Fermat prime** - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $257$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Fermat number**(P. de Fermat, 1640) - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Fermat number**(P. de Fermat, 1640) - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $7$: Patterns in Numbers: Gauss