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A343913 Positive integers m such that 2*m^2 - 1 = x^4 + y^4 for some nonnegative integers x and y with |x - y| > 1.

%I A343913 #36 Nov 04 2023 13:27:45
%S A343913 71,347,1193,2139,2709,17823,18337,26057,32847,34037,65793,87519,
%T A343913 159541,245573,383037,421957,489731,520547,574841,800589,1291333,
%U A343913 2010341,2113003,2990187,4528667,7430553,8284063,8402417,8520567,9220519,9865989,10621507,11961043,12335203,16405581,17648561,22224647,22918853,24171273
%N A343913 Positive integers m such that 2*m^2 - 1 = x^4 + y^4 for some nonnegative integers x and y with |x - y| > 1.
%C A343913 Conjecture: The sequence has infinitely many terms.
%C A343913 Clearly all the terms must be odd and not divisible by 5. Note also that 2*(n^2+n+1)^2 - 1 = n^4 + (n+1)^4.
%C A343913 See also A343917 for a similar conjecture.
%H A343913 Robert Israel, <a href="/A343913/b343913.txt">Table of n, a(n) for n = 1..112</a> (all terms < 10^10; first 53 terms from Zhi-Wei Sun)
%e A343913 a(1) = 71, and 2*71^2 - 1 = 10^4 + 3^4 with |10 - 3| > 1.
%e A343913 a(53) = 99532937, and 2*99532937^2 - 1 = 19813611095691937 = 11337^4 + 7576^4 with |11337 - 7576| > 1.
%p A343913 N:= 10^18: # for all terms <= sqrt(N)
%p A343913 R:= {}: count:= 0:
%p A343913 for x from 1 while 2*x^4 < 2*N-1 do
%p A343913   for y from x+3 by 2 do
%p A343913     v:= (x^4 + y^4 + 1)/2;
%p A343913     if v > N then break fi;
%p A343913     if issqr(v) then
%p A343913       m:= sqrt(v);
%p A343913       if not member(m,R) then
%p A343913          count:= count+1; R:= R union {m};
%p A343913       fi fi
%p A343913 od od:
%p A343913 sort(convert(R,list)); # _Robert Israel_, May 04 2021
%t A343913 QQ[n_]:=IntegerQ[n^(1/4)];
%t A343913 n=0;Do[Do[If[QQ[2*m^2-1-(2x)^4]&&Abs[2x-(2*m^2-1-(2x)^4)^(1/4)]>1,n=n+1;Print[n," ",m];Goto[aa]],{x,0,((2m^2-1)^(1/4))/2}];Label[aa],{m,1,25000000}]
%Y A343913 Cf. A000290, A000583, A003336, A056220, A343917.
%K A343913 nonn
%O A343913 1,1
%A A343913 _Zhi-Wei Sun_, May 03 2021