## Wednesday, June 30, 2021

### [rivzkfzp] power tower Fermat numbers

the 16th Fermat number is 2^65536 + 1 = 2^2^16 + 1 = 2^2^2^2^2 + 1 = 2^^5 + 1 = (2 -> 5 -> 2) + 1.  (this is a use of Conway's chained arrow notation in an application outside of numbers just for the sake of being huge.)  it has two known prime factors, 1575*2^19+1 = 825753601 and 180227048850079840107*2^20+1 = 188981757975021318420037633 .  the remaining 19694-digit cofactor is composite.

the 64th Fermat number is 2^2^2^6 + 1 .  we know of 1 prime factor, 17853639 * 2^67 + 1 = 2634732075339197803231444993 .  the cofactor is too large to be tested for primality.

36986355 * 2^258 + 1 is the one known factor of the 256th Fermat number 2^2^2^2^3 + 1 .

we do not know any factors of the 65536th Fermat number 2^2^2^2^2^2 + 1 = 2^^6 + 1 = (2 -> 6 -> 2) + 1.  here is some Pari/GP code we ran, testing multipliers k up to 1192240, a few weeks of computing.  we started at 100000 because the website said multipliers below that had already been tested.

we did not find any factors.

m=2^16; for(k=100000, 1192240, j=k*2^(m+2)+1; fail=0; forprime(p=3, 5600000, if(j%p==0, fail=1; break)); if(!fail, print("check ", k); if(0==(1+Mod(2, j)^2^m), print("====found ", i); break)))

previously, on the optimal amount of trial division, used in the prime limit 5600000 above.  we only search for prime factors of the Fermat number because if there is a composite factor of the proper form, its prime factors would have been found earlier.

all positive results above from http://www.prothsearch.com/fermat.html .