A157950 Primes p such that p^8 + 2^8 is prime.

13, 137, 223, 331, 389, 491, 563, 647, 701, 773, 797, 1063, 1181, 1531, 1579, 1811, 2027, 2087, 2269, 2333, 2393, 2617, 2687, 2699, 2857, 3313, 3467, 3623, 3637, 3691, 3739, 3761, 3863, 3877, 4133, 4201, 4283, 4297, 4877, 5023, 5839, 5897, 6043, 6053

Offset: 1

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 10 2009

Keywords

• nonn

Comments

17 divides p^8 + 2^8 if k is odd and p = 17k +- 6, 17k +- 10, 17k +- 12, 17k +- 14, so only 8 integers p in each interval of length 34 need to be tested for the primality of p and of p^8 + 2^8: those of the forms p = 17k +- 2 (which yield terms 223, 389, 491, 563, 797, 1579, 3313, 3623, 3691, ...), p = 17k +- 4 (which yield terms 13, 701, 2027, 2087, 2333, 2393, 2699, ...), p = 17k +-8 (which yield terms 331, 773, 1063, 1181, 1811, 2269, ...), and p = 17k +-16 (which yield terms 137, 647, 1531, 2617, 2687, 2857, 3467, 3637, ...).

It is conjectured that this sequence is infinite.

Examples

			n=11: 11^8 + 2^8 = 214359137 = 17 * 241 * 52321, not prime, so 11 is not a term;
n=13: 13^8 + 2^8 = 815730977 is prime, so 13 is a term.
		

References

• Leonard E. Dickson, History of the Theory of Numbers.
• Richard Guy, Unsolved Problems in Number Theory.

Crossrefs

Cf. A062324, A157764.

Programs

Maple

a := proc (n) if isprime(ithprime(n)^8+256) = true then ithprime(n) else end if end proc: seq(a(n), n = 1 .. 900); # Emeric Deutsch, Mar 14 2009

Extensions

Definition corrected by Emeric Deutsch, Mar 14 2009

Extended by Emeric Deutsch, Mar 14 2009

Edited by Jon E. Schoenfield, Jan 29 2019