# Definition:Fermat Number/Historical Note

## Historical Note on Fermat Number

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

- 1906: J.C. Morehead:
*Note on the factors of Fermat's numbers*(*Bull. Amer. Math. Soc.***Vol. 12**: pp. 449 – 451)

- 1909: J.C. Morehead and A.E. Western:
*Note on Fermat's numbers*(*Bull. Amer. Math. Soc.***Vol. 18**: pp. 1 – 6)

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{IV}$: The Prince of Amateurs

- 1971: Michael A. Morrison and John Brillhart:
*The factorization of $F_7$*(*Bull. Amer. Math. Soc.***Vol. 77**: p. 264)

- 1981: Richard P. Brent and John M. Pollard:
*Factorization of the Eighth Fermat Number*(*Math. Comp.***Vol. 36**: pp. 627 – 630) www.jstor.org/stable/2007666

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $257$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $4,294,967,297$ - 1989: Paulo Ribenboim:
*The Book of Prime Number Records*(2nd ed.) - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $257$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $4,294,967,297$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $F_{23471}$