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%I #20 Sep 08 2022 08:46:19
%S 3,17,577,665857
%N Primes p such that (p^2 - 1) / 2 is a square (A000290).
%C Corresponding values of squares: 4, 144, 166464, 221682772224.
%C Subsequence of A257553.
%C Conjecture: sequence is finite.
%C Numbers k such that (k^2 - 1) / 2 is a square are given by A001541, of which the only prime terms are 3, 17, 577, and 665857 (see _Alexander Adamchuk_'s Nov 24 2006 Comments entry there), so a(4) = 665857 is the last term of this sequence. - _Jon E. Schoenfield_, Nov 20 2017
%e Number 3 is in the sequence because (3^2 - 1) / 2 = 4 (square).
%t Select[Prime[Range[55000]],IntegerQ[Sqrt[(#^2-1)/2]]&] (* _Harvey P. Dale_, Mar 10 2019 *)
%o (Magma) [n: n in [3..1000000] | IsPrime(n) and IsSquare((n^2-1) / 2)]
%Y Cf. A088165 (primes p such that (p^2 + 1) / 2 is a square).
%Y Cf. A000290, A002315, A257553.
%K nonn,fini,full
%O 1,1
%A _Jaroslav Krizek_, Sep 12 2017