A378143 a(n) is the smallest prime of the form (2*p)^(2^n) + 1 for some prime p.

5, 17, 257, 65537, 808551180810136214718004658177, 9807585394417153072393128067370344132933540474708183331242417216238928121991128579833857

Offset: 0

Author

Juri-Stepan Gerasimov, Nov 17 2024

Keywords

• nonn

Comments

If p = 2, then a(n) is the Fermat prime.

Conjecture: the last digit of each value of a(n), where n >= 1, is 7.

The conjecture is equivalent to the claim that a(n) is not 10^(2^n) + 1 for any n, which in turn is equivalent to the claim that, if 10^(2^n) + 1 is prime, then either 4^(2^n) + 1 or 6^(2^n) + 1 is prime. - Charles R Greathouse IV, Nov 17 2024

Crossrefs

Primes p such that (2*p)^(2^k) + 1 is prime: A005384 (k = 0), A052291 (k = 1), A378146 (k = 2).

Cf. A019434, A222008, A286678, A378134.