Definition:Fermat Number/Historical Note

Historical Note on Fermat Number
In $1640$, wrote to  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, 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, who discovered the prime decomposition of the $6$th Fermat number $F_5$.

In $1877$, 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 $F_{23}$.

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 $1909$, and  reported in  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, who found the divisor $5 \times 2^{75} + 1$ in $1906$.

The prime factors of $F_7$ were finally discovered by and  in $1970$:
 * $F_7 = \left({116 \, 503 \, 103 \, 764 \, 643 \times 2^9 + 1}\right) \left({11 \, 141 \, 971 \, 095 \, 088 \, 142 \, 685 \times 2^9 + 1}\right)$

One of the divisors of $F_8$ was found by and  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 $F_{23 \, 471}$