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%I #32 Apr 11 2016 23:11:09
%S 17,35,51,73,105,145,195,233,273,289,291,339,451,465,521,577,579,585,
%T 611,675,723,777,801,809,819,899,915,969,1043,1059,1097,1153,1155,
%U 1185,1225,1281,1313,1347,1353,1395,1545,1601,1603,1665,1683,1731,1745,1763,1801,1873,1923,1961,1971,2019
%N Integers n such that n^2-1 and n^2 are the sum of two nonzero squares.
%C Corresponding n^2 values are 289, 1225, 2601, 5329, 11025, 21025, 38025, 54289, 74529, 83521, 84681, ...
%C If n is in this sequence, so is n^2. Proof:
%C n is a term of this sequence if and only if n^2 = a^2 + b^2 and n^2-1 = c^2 + d^2 for a,b,c,d are nonzero integers.
%C If n^2 = a^2 + b^2, then n^4 = (n^2)*(n^2) = (n^2)*(a^2 + b^2) = (n*a)^2 + (n*b)^2.
%C If n^2-1 = c^2 + d^2, then n^4-1 = (n^2-1)*(n^2+1) = (c^2 + d^2)*(n^2+1) = (c*n + d)^2 + (d*n - c)^2. Note that (d*n - c) > 0 because of definition of n.
%C Since both (n^2)^2 and (n^2)^2-1 are the sum of two nonzero squares, n^2 must be a term.
%C With repeating of same procedure, it can be seen that if n is in this sequence, so is n^(2^k), for k >= 0.
%e 17 is a term because 17^2 = 8^2 + 15^2 and 17^2 - 1 = 12^2 + 12^2.
%t fQ[n_] := Length[PowersRepresentations[n, 2, 2] /. {0, _} -> Nothing] > 0; Select[Range@ 2020, And[fQ[#^2 - 1], fQ[#^2]] &] (* _Michael De Vlieger_, Apr 10 2016 *)
%o (PARI) isA000404(n) = {for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2))}
%o for(n=1, 1e4, if(isA000404(n^2-1) && isA000404(n^2), print1(n, ", ")))
%Y Cf. A000404, A064716.
%K nonn
%O 1,1
%A _Altug Alkan_, Apr 09 2016
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