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A343917 Positive integers m with 2*m^2 - 2^4 = x^4 + y^4 for some nonnegative integers x and y with |x-y| > 2.

%I A343917 #19 May 04 2021 09:04:33
%S A343917 284,1388,2139,4772,8556,8971,10836,21163,28847,45707,54507,71292,
%T A343917 73348,95127,101503,104228,131388,136148,263172,350076,638164,982292,
%U A343917 1532148,1687828,1705407,1958924,2082188,2299364,2360347,2728379,3202356,4042799,5046771,5165332,5235323,5560627,7191079,7740547,8041364
%N A343917 Positive integers m with 2*m^2 - 2^4 = x^4 + y^4 for some nonnegative integers x and y with |x-y| > 2.
%C A343917 Conjecture: The sequence has infinitely many terms.
%C A343917 It is easy to see that no term is divisible by 5. In the b-file we list all the 62 terms not exceeding 10^8.
%C A343917 Note that 2*(n^2+3)^2 - 2^4 = (n+1)^4 + (n-1)^4 with (n+1) - (n-1) = 2. This implies that any integer n > 4 can be written as  x + y + 2^(z-1) with x,y,z positive integers such that x^4 + y^4 + (2^z)^4 is twice a square.
%C A343917 See also A343913 for a similar conjecture.
%H A343917 Zhi-Wei Sun, <a href="/A343917/b343917.txt">Table of n, a(n) for n = 1..62</a>
%e A343917 a(1) = 284, and 2*284^2 - 2^4 = 20^4 + 6^4 with |20-6| > 2.
%e A343917 a(62) = 97077407, and 2*97077407^2 - 2^4 = 18848045899687282 = 11563^4 + 5583^4 with |11563-5583| > 2.
%t A343917 QQ[n_]:=IntegerQ[n^(1/4)];
%t A343917 n=0;Do[Do[If[QQ[2*m^2-16-x^4]&&(2*m^2-16-x^4)^(1/4)-x>2,n=n+1;Print[n," ",m];Goto[aa]],{x,0,(m^2-8)^(1/4)}];Label[aa],{m,3,8041364}]
%Y A343917 Cf. A000290, A000583, A003336, A230121, A230747, A340274, A343862, A343897, A343913.
%K A343917 nonn
%O A343917 1,1
%A A343917 _Zhi-Wei Sun_, May 04 2021