A080208 a(n) is the least k such that the generalized Fermat number (k+1)^(2^n) + k^(2^n) is prime.

1, 1, 1, 1, 1, 8, 95, 31, 85, 59, 1078, 754, 311, 3508, 1828, 49957, 22844

Offset: 0

Author

T. D. Noe, Feb 10 2003

Keywords

• hard
• more
• nonn

Comments

The first five terms correspond to the five known Fermat primes. The sequence A078902 lists some of the generalized Fermat primes. Bjorn and Riesel examined generalized Fermat numbers for k <= 11 and n <= 999. The sequence A080134 lists the conjectured number of primes for each k.

For n >= 10, a(n) yields a probable prime. a(13) was found by Henri Lifchitz. It is known that a(14) > 1000.

Examples

			a(5) = 8 because (k+1)^32 + k^32 is prime for k = 8 and composite for k < 8.
		

Links

• T. D. Noe, Table of generalized Fermat primes of the form (k+1)^2^m + k^2^m
• Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
• Eric Weisstein's World of Mathematics, Generalized Fermat Number

Crossrefs

Cf. A019434, A078902, A080134, A153504, A152913, A194185, A253633.

Formula

a(n) = A253633(n) - 1.

Extensions

a(14)-a(15) from Jeppe Stig Nielsen, Nov 27 2020

a(16) by Kellen Shenton communicated by Jeppe Stig Nielsen, May 19 2023