A291944 a(n) is the least A for which there exists B with 0 < B < A so that A^(2^n) + B^(2^n) is prime.

2, 2, 2, 2, 2, 9, 11, 27, 14, 13, 47, 22, 53, 72, 216, 260, 124, 1196, 200

Offset: 0

Author

Jeppe Stig Nielsen, Mar 09 2018

Keywords

• nonn
• hard
• more

Comments

A^(2^n) + B^(2^n) is called an (extended) generalized Fermat prime, and often denoted F_n(A, B); or xGF(n, A, B).

If we require B=1, we get A056993. Therefore a(n) <= A056993(n).

Examples

			a(10)=47 corresponds to the prime number 47^1024 + 26^1024, the smallest prime number of the form A^1024 + B^1024 (or more precisely, it minimizes A).
a(14)=216 corresponds to the prime number 216^16384 + 109^16384, a 38248-decimal digit PRP, the smallest prime number of the form A^16384 + B^16384. - _Serge Batalov_, Mar 16 2018
		

Links

• Jeppe Stig Nielsen, List of [n, A, B] tuples for this sequence.
• Chris K. Caldwell, The Prime Database: 72^8192 + 43^8192 (related to a(13)).
• Henri Lifchitz & Renaud Lifchitz, 1196^131072+595^131072, a(17).
• Henri Lifchitz & Renaud Lifchitz, 200^262144+119^262144, a(18).

Crossrefs

Cf. A056993, A253633, A111635.

Programs

Mathematica

f[n_] := Monitor[ Block[{a = 2, b}, While[a < Infinity, b = 1 +Mod[a, 2]; While[b < a, If[ PrimeQ[a^2^n + b^2^n], Goto[fini]]; b+=2]; a++]; Label[fini]; {a, b}], {a, b}]; Array[f, 14, 0] (* Robert G. Wilson v, Mar 10 2018 *)

PARI

for(n=0,30,for(a=2,10^100,forstep(b=(a % 2)+1,a-1,2,if(ispseudoprime(a^(2^n)+b^(2^n)),print1(a,", ");next(3)))))

Extensions

a(14) = 216 (and B = 109) from Serge Batalov, Mar 16 2018

a(15) = 260 (and B = 179) from Serge Batalov, Mar 16 2018

a(16) = 124 (and B = 57) from Serge Batalov, Mar 16 2018

a(17) = 1196 (and B = 595) from Kellen Shenton, Aug 10 2022

a(18) = 200 (and B = 119) from Kellen Shenton, Aug 27 2022