%I #14 Sep 08 2022 08:45:44
%S 3,23,47,53,67,73,89,101,103,109,151,157,179,229,521,557,569,619,661,
%T 821,977,1013,1087,1129,1277,1321,1451,1559,1607,1627,1741,1867,1871,
%U 1949,2137,2389,2441,2797,3271,3313,3643,3677,3769,3847,4001,4027,4133
%N Primes p such that p^4 + 5^4 + 3^4 is prime.
%C For primes p, q, r the sum p^4 + q^4 + r^4 can be prime only if at least one of p, q, r equals 3. This sequence is the special case q = 5, r = 3.
%C It is conjectured that the sequence is infinite.
%C There are twin prime (101, 103) and other consecutive primes (151, 157; 1867, 1871) in the sequence.
%e p = 3: 3^4 + 5^4 + 3^4 = 787 is prime, so 3 is in the sequence.
%e p = 5: 5^4 + 5^4 + 3^4 = 1331 = 11^3, so 5 is not in the sequence.
%e p = 101: 101^4 + 5^4 + 3^4 = 104061107 is prime, so 101 is in the sequence.
%e p = 103: 103^4 + 5^4 + 3^4 = 112551587 is prime, so 103 is in the sequence.
%t With[{c=5^4+3^4},Select[Prime[Range[600]],PrimeQ[#^4+c]&]] (* _Harvey P. Dale_, Aug 14 2011 *)
%o (Magma) [p: p in PrimesUpTo(5000)|IsPrime(p^4+706)] // _Vincenzo Librandi_, Dec 18 2010
%o (PARI) is(n)=isprime(n) && isprime(n^4+706) \\ _Charles R Greathouse IV_, Jun 07 2016
%Y Cf. A158979, A159829, A160031.
%K easy,nonn
%O 1,1
%A Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 30 2009
%E Edited, 1607 inserted and extended beyond 3643 by _Klaus Brockhaus_, May 03 2009