Fermat Numbers

A Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a positive integer of the form: $$ \text{F}_n=2^{2^n}+1 $$ where n is a non-negative integer. The first few Fermat numbers are:

• \(\text{F}_0=3\)
• \(\text{F}_1=5\)
• \(\text{F}_2=17 \)
• \(\text{F}_3=257\)
• \(\text{F}_4=65537\)
• \(\text{F}_5=4294967297\)
• \(\text{F}_6=18446744073709551617\)
• \(\text{F}_7=340282366920938463463374607431768211457\)

The numbers form the initial members of sequence A000215 in the OEIS. The only known Fermat primes are the first five members of this sequence (3, 5, 17, 257 and 65537).

The number associated with my diurnal age today, \( \textbf{28289}\), is a prime and oddly it has a connection to these Fermat numbers via its membership in OEIS A229856:

   A229856: primes of the form \(384 \times k + 257\) where \(k \geq 0\)

The initial members are:

257, 641, 1409, 3329, 4481, 7937, 9473, 9857, 11393, 11777, 12161, 13313, 13697, 14081, 15233, 16001, 17921, 19073, 19457, 19841, 21377, 23297, 25601, 28289, 30593, 30977, 35201, 35969, 36353, 37889, 38273, 39041

It's stated in the comments to this OEIS sequence that:

Every Fermat number greater than 257 has a prime factor of the form \(384 \times k + 257, k > 0\).

Let's test this out with the initial Fermat numbers greater than 257. Now \( \text{F}_4=65537\) is a prime and so it is its own prime factor and we find that:$$65537 = 384 \times 170 + 257$$The next Fermat numbers listed above are all composite so we need to check their prime factors:$$ \begin{align} \text{F}_5=4294967297 &= 641 \times 6700417 \\ 641 &= 384 \times 1 + 257 \end{align}$$ $$ \begin{align} \text{F}_6=18446744073709,551,617 &=274177 × 67280421310721\\ 67280421310721 &= 384 \times 175209430496+257 \end{align}$$I won't attempt \( \text{F}_7 \) as it's too lengthy but the point is made. There's an explanation that Gemini provided as to why at least one prime factor of a Fermat number must have this form. You can check that out here.