A158361 Primes p with property that Q = p^4 + 2^4 is prime.

3, 5, 7, 11, 17, 19, 23, 37, 41, 59, 61, 71, 79, 97, 131, 139, 179, 223, 227, 229, 241, 283, 313, 317, 359, 367, 379, 383, 389, 439, 449, 461, 487, 503, 521, 569, 593, 617, 619, 631, 661, 683, 709, 733, 811, 821, 853, 911, 977, 1049, 1061, 1063, 1069, 1091, 1093, 1117

Offset: 1

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 17 2009

Keywords

• nonn
• easy

Comments

Q is always congruent to 1 (mod 4).

Q is divisible by 17 if p is congruent to 1, 4, 13, or 16 (mod 17).

It is conjectured that sequence a(n) is infinite.

Q is in A094479. - Zak Seidov, Jul 08 2020

Examples

			3 is in the sequence since for p=3: p^4+2^4 = 3^4+16 = 97 is prime.
29 is not in the sequence since 29^4+2^4 = 707297 = 73 x 9689 is not prime.
		

References

• Leonard E. Dickson, History of the Theory of numbers, vol. I, Dover Publications 2005
• Richard Guy, "Unsolved Problems in Number Theory"

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A062324, A094479, A157950, A157764.

Programs

Magma

[p: p in PrimesUpTo(2000) | IsPrime(p^4+16)]; // Vincenzo Librandi, Jun 18 2014

Mathematica

Select[Range[10^3], PrimeQ[#] && PrimeQ[#^4 + 16] &] (* Vincenzo Librandi, Jun 18 2014 *)
Select[Prime[Range[200]],PrimeQ[#^4+16]&] (* Harvey P. Dale, Jun 23 2014 *)

PARI

isA158361(n) = isprime(n) && isprime(n^4+16)

Extensions

Corrected and edited by Michael B. Porter, Dec 17 2009