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%I #37 Sep 08 2022 08:46:12
%S -4,-3,-2,0,1,2,8,10,18,112,1272
%N Integers n such that n*(n + 2)*(n + 4) + 1 is a perfect square.
%C This sequence is finite as there are finitely many integer solutions to the elliptic curve y^2 = x(x + 2)(x + 4) + 1 = x^3 + 6x^2 + 8x + 1. The x values of the integer solutions are {-4, -3, -2, 0, 1, 2, 8, 10, 18, 112, 1272}. This equation has more integer and natural number solutions than the equation that defines sequence A121234.
%e 1 * 3 * 5 + 1 = 16 = 4^2, so 4 is in the sequence.
%e 2 * 4 * 6 + 1 = 49 = 7^2, so 2 is in the sequence.
%e 3 * 5 * 7 + 1 = 106 = 2 * 53, so 3 is not in the sequence.
%t Select[Range[-10, 100], IntegerQ[Sqrt[#(# + 2)(# + 4) + 1]] &] (* _Alonso del Arte_, Jun 12 2015 *)
%o (Magma) P<n> := PolynomialRing(Integers()); {x: x in Sort([ p[1] : p in IntegralPoints(EllipticCurve(n^3 + 6*n^2 + 8*n + 1)) ])};
%o (SageMath) [i[0] for i in EllipticCurve([0, 6, 0, 8, 1]).integral_points()] # _Seiichi Manyama_, Aug 26 2019
%Y Cf. A121234.
%K sign,fini,full
%O 1,1
%A _Morris Neene_, Jun 12 2015
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