A217993 Smallest k such that k^(2^n) + 1 and (k+2)^(2^n) + 1 are both prime.

2, 2, 2, 2, 74, 112, 2162, 63738, 13220, 54808, 3656570, 6992032, 125440, 103859114, 56414914, 87888966

Offset: 0

Author

Michel Lagneau, Oct 17 2012

Keywords

• nonn
• hard
• more

Comments

a(15)=87888966 but a(14) is unknown. - Jeppe Stig Nielsen, Mar 17 2018

The prime pair related to a(14) was found four days ago, and today double checking has proved that they are indeed the first occurrence for n=14. - Jeppe Stig Nielsen, May 02 2018

Examples

			a(0) = 2 because 2^1+1 = 3 and 4^1+1 = 5 are prime;
a(1) = 2 because 2^2+1 = 5  and 4^2+1 = 17 are prime;
a(2) = 2 because 2^4+1 = 17  and 4^4+1 = 257 are prime;
a(3) = 2 because  2^8+1 = 257 and 4^8+1 = 65537 are prime.
		

Crossrefs

Cf. A006313, A006314, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A118539.

Programs

Maple

for n from 0  to 5 do:ii:=0:for k from 2 by 2 to 10000 while(ii=0) do:if type(k^(2^n)+1,prime)=true and type((k+2)^(2^n)+1,prime)=true then ii:=1: printf ( "%d %d \n",n,k):else fi:od:od:

Formula

a(n) = A118539(n)-1. - Jeppe Stig Nielsen, Feb 27 2016

Extensions

a(13) from Jeppe Stig Nielsen, Mar 17 2018

a(14) and a(15) from Jeppe Stig Nielsen, May 02 2018