[seqfan] Re: ^2 + 2^

[israel at math.ubc.ca]

On Feb 22 2016, 2stepan at rambler.ru wrote:

> Dear SeqFans: primes p such that p^2 + 2^(2^k) is not prime for all k: 2, 
> 23, 43, 53, 83, 107, 113, 127, 157, 173, 197, 223, 227, 257, 263, 283, 
> 337, 353, 367, 383, 397,433,443, 457, 467,523, 557, 563, 587,617, 643, 
> 647, 653, 673, 683,727, 733, 757, 773, 797, 857, 863, 877, 887, 907, 937, 
> 947, 977, 1013, 1033, 1063, 1097, ... This sequence is correct? Best, JSG

Well, certainly if p == 2 or 3 mod 5, p^2 == 4 mod 5 and then p^2 + 2^(2^k) 
is divisible by 5 for k >= 2, so if p^2 + 2 = p^2 + 2^(2^0) and p^2 + 4 = 
p^2 + 2^(2^1) are not prime, none of the others will be. If p == 1 or 4 mod 
5, I don't know how you would rule out some p^2 + 2^(2^k) being prime. Thus 
what about p = 59?

Cheers,
Robert