%I #13 Feb 16 2025 08:33:03

%S 3,5,17,257,65537,185302018885184100000000000000000000000000000001

%N Smallest generalized Fermat prime of the form a^(2^n) + 1, where base a>1 is an integer; or -1 if no such prime exists.

%C First 5 terms {3, 5, 17, 257, 65537} = A019434 are the Fermat primes of the form 2^(2^n) + 1. Note that for all currently known a(n) up to n = 17 last digit is 7 or 1 (except a(0) = 3 and a(1) = 5). Corresponding least bases a>1 such that a^(2^n) + 1 is prime are listed in A056993.

%C The last-digit behavior clearly continues since, for any a, we have that a^(2^2) will be either 0 or 1 modulo 5. So for n >= 2, a(n) is 1 or 2 modulo 5, and odd. - _Jeppe Stig Nielsen_, Nov 16 2020

%H Charles R Greathouse IV, <a href="/A123599/b123599.txt">Table of n, a(n) for n = 0..9</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GeneralizedFermatNumber.html">Generalized Fermat Number</a>.

%t Do[f=Min[Select[ Table[ i^(2^n) + 1, {i, 2, 500} ],PrimeQ]];Print[{n,f}],{n,0,9}]

%Y Cf. A000215, A019434, A056993.

%Y Cf. A006093, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002.

%K nonn

%O 0,1

%A _Alexander Adamchuk_, Nov 14 2006