A158979 - OEIS

%I #14 Aug 09 2023 19:02:13

%S 2,3,4,5,8,7,10,9,10,13,16,13,14,15,22,17,20,23,24,29,38,29,26,41,26,

%T 27,28,33,34,37,32,37,34,35,52,37,38,39,46,41,50,53,44,47,58,55,50,49,

%U 60,61,62,61,56,55,58,59,68,61,62,73,66,77,64,67,84,71

%N a(n) is the smallest number > n such that n^4 + a(n)^4 is prime.

%C For exponent 2 instead of 4 see A089489: Pythagorean triple has a prime hypotenuse.

%C Corresponding sequences with odd exponent u are impossible: x^u + y^u has factor x+y.

%C a(2k-1) is even, a(2k) is odd, a(n)-n is odd.

%C Conjecture: a(n) exists for all n, i.e., the sequence is well-defined and infinite.

%C Conjecture: a(n)-n = 1 for infinitely many n.

%C The largest value of a(n)-n for n <= 100 occurs at n = 90: 121-90 = 31.

%C a(n)-n = 1 for 35 values of n <= 100.

%H Harvey P. Dale, <a href="/A158979/b158979.txt">Table of n, a(n) for n = 1..1000</a>

%e 1^4 + 2^4 = 17 is prime, so a(1) = 2.

%e 2^4 + 3^4 = 97 is prime, so a(2) = 3.

%e 5^4 + 6^4 = 1921 = 17*113, 5^4 + 7^4 = 3026 = 2*17*89, 5^4 + 8^4 = 4721 is prime, so a(5) = 8.

%t sn[n_]:=Module[{k=n+1,n4=n^4},While[CompositeQ[n4+k^4],k++];k]; Array[sn,80] (* _Harvey P. Dale_, Aug 09 2023 *)

%o (Magma) S:=[]; for n in [1..72] do q:=n^4; k:=n+1; while not IsPrime(q+k^4) do k+:=1; end while; Append(~S, k); end for; S; // _Klaus Brockhaus_, Apr 12 2009

%Y Cf. A089489.

%K easy,nonn

%O 1,1

%A Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 01 2009

%E Edited and entries verified by _Klaus Brockhaus_, Apr 12 2009

%E Corrected by _Harvey P. Dale_, Aug 09 2023