A274002 Primes p of the form (q-1)^2+1 that are a divisor of 4^(q-1)-1 where q is prime.

5, 17, 257, 65537, 3497539601

Offset: 1

Author

Jaroslav Krizek, Jun 06 2016

Keywords

• nonn
• more

Comments

Corresponding values of primes q: 3, 5, 17, 257, 59141, ...

The first 4 known Fermat primes > 3 from A019434 are in this sequence.

Conjecture: also primes p of the form (q-1)^2+1, where q = prime, that are a divisor of (4^k)^(q-1)-1 for all k>=0. Example: 17 = (5-1)^2+1 is a term because 5 is prime and divides (4^k)^(5-1)-1 for all k>=0: 0/17 = 0 (k=0); 255/17 = 15 (k=1); 65535/17 = 3855 (k=2); 16777215/17 = 986895 (k=3); 4294967295/17 = 252645135 (k=4); 1099511627775/17 = 64677154575 (k=5); ...

Subsequence of A274000.

Examples

			17 = (5-1)^2+1 is a term because 17 divides 4^(5-1)-1; 255/17 = 15.
		

Crossrefs

Cf. A019434, A273999, A274000.

Programs

PARI

listp(nn) = {forprime(p=2, nn, if (isprime(q=(p-1)^2 + 1) &&  (Mod(4, q)^(p-1) == 1), print1(q, ", ")););} \\ Michel Marcus, Jun 08 2016