A343121 a(n) is the least A for which there exists B with 0 < B < A so that A^(2^k) + B^(2^k) is prime for k = 0, 1, ..., n.

2, 2, 2, 2, 2, 2669, 34559, 26507494, 3242781025

Offset: 0

Author

Jeppe Stig Nielsen, Apr 05 2021

Keywords

• nonn
• more
• hard

Comments

For n < 5, the corresponding primes are Fermat primes, for higher n so-called generalized Fermat primes.

Examples

			For n=5, the six numbers 2669 + 720, 2669^2 + 720^2, 2669^4 + 720^4, 2669^8 + 720^8, 2669^16 + 720^16, and 2669^32 + 720^32 are all prime, and (A,B) = (2669,720) is the least pair with this property, so a(5)=2669.
For n=6, (A,B) = (34559,29000).
For n=7, (A,B) = (26507494,6329559).
For n=8, (A,B) = (3242781025,1554825312).
		

Links

• Yves Gallot, xgfp8, software for calculating this sequence.

Crossrefs

Cf. A090872, A111635, A291944, A343120.

Programs

PARI

a(n)=for(A=1, oo, for(B=1, A-1, for(k=0, n, !ispseudoprime(A^(2^k)+B^(2^k)) && next(2)); return(A)))

Extensions

a(7) from Kellen Shenton, May 28 2022

a(8) from Kellen Shenton, Aug 27 2022