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Primes p such that p^4 + 13^4 + 3^4 is prime.
1

%I #10 Sep 08 2022 08:45:44

%S 3,11,13,17,31,41,43,53,83,127,167,181,193,211,241,311,337,349,421,

%T 431,487,521,557,613,617,647,701,769,811,857,953,1021,1151,1249,1289,

%U 1303,1373,1453,1459,1471,1523,1553,1567,1579,1613,1663,1669,1747,1823,1831

%N Primes p such that p^4 + 13^4 + 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 = 13, r = 3.

%C It is conjectured that the sequence is infinite.

%C There are prime twins (11, 13) and other consecutive primes (421, 431; 1823, 1831) in the sequence.

%H Harvey P. Dale, <a href="/A160025/b160025.txt">Table of n, a(n) for n = 1..1000</a>

%e p = 3: 3^4 + 13^4 + 3^4 = 28723 is prime, so 3 is in the sequence.

%e p = 5: 5^4 + 13^4 + 3^4 = 29267 = 7*37*113, so 5 is not in the sequence.

%e p = 17: 17^4 + 13^4 + 3^4 = 112163 is prime, so 17 is in the sequence.

%e p = 83: 83^4 + 13^4 + 3^4 = 47486963 is prime, so 83 is in the sequence.

%t Select[Prime[Range[400]],PrimeQ[#^4+28642]&] (* _Harvey P. Dale_, Dec 14 2011 *)

%o (Magma) [ p: p in PrimesUpTo(1840) | IsPrime(p^4+28642) ]; // _Klaus Brockhaus_, May 03 2009

%Y Cf. A158979, A159829, A160022.

%K easy,nonn

%O 1,1

%A Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 30 2009

%E Edited and extended beyond 857 by _Klaus Brockhaus_, May 03 2009