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

1,1

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.

References

Leonard E. Dickson, History of the Theory of Numbers.

Richard Guy, Unsolved Problems in Number Theory.

Links

Table of n, a(n) for n=1..44.

Example

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.

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

Crossrefs

Cf. A062324, A157764.

Sequence in context: A078795 A123299 A142017 * A249444 A046278 A016205

Adjacent sequences: A157947 A157948 A157949 * A157951 A157952 A157953

Keyword

nonn

Author

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

Status

approved