A253633 a(n) is the least positive integer b such that b^(2^n) + (b-1)^(2^n) is prime.

2, 2, 2, 2, 2, 9, 96, 32, 86, 60, 1079, 755, 312, 3509, 1829, 49958, 22845

Offset: 0

Author

Jeppe Stig Nielsen, Jan 07 2015

Keywords

• nonn
• hard
• more

Comments

When a(n) is 2, the corresponding prime is a Fermat prime, otherwise it is a so-called extended generalized Fermat prime sometimes denoted xGF(n, b, b-1) or similar.

Examples

			For n = 5, 2^5 = 32 is the exponent.  The numbers 1^32 + 0^32, 2^32 + 1^32, ..., 8^32 + 7^32 are not prime, but 9^32 + 8^32 is prime, so a(5) = 9. - _Michael B. Porter_, Mar 28 2018
		

Links

• Henri Lifchitz & Renaud Lifchitz, PRP Top Records, search for x^16384+y^16384, related to a(14).
• Henri Lifchitz & Renaud Lifchitz, 49958^32768+49957^32768, a(15).
• Henri Lifchitz & Renaud Lifchitz, 22845^65536+22844^65536, a(16).

Crossrefs

Cf. A056993, A080208.

Programs

PARI

a(n)=for(b=2,10^10,if(ispseudoprime(b^(2^n)+(b-1)^(2^n)),return(b)))

Formula

a(n) = A080208(n) + 1.

Extensions

a(13) from Jeppe Stig Nielsen, Mar 27 2018

a(14) found by Henri Lifchitz in 2007, from Jeppe Stig Nielsen, Apr 17 2018

a(15) found by Kellen Shenton, from Jeppe Stig Nielsen, Nov 27 2020

a(16) found by Kellen Shenton, from Jeppe Stig Nielsen, Mar 31 2021