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 A121234 Nonnegative integers k such that k*(k+1)*(k+2)+1 is a square. 2

%I

%S 0,2,4,55

%N Nonnegative integers k such that k*(k+1)*(k+2)+1 is a square.

%C The sequence is finite by Thue's theorem n(n + 1)(n + 2) + 1 = n^3 + 3*n^2 + 2*n + 1. The set of k values of integral solutions to the elliptic curve y^2 = n^3 + 3*n^2 + 2*n + 1 (see Magma program) is { -2, -1, 0, 2, 4, 55 }. So the sequence is complete. - _Mohamed Bouhamida_, Nov 29 2007

%e 2 * 3 * 4 + 1 = 25 = 5^2, so 2 is in the sequence.

%e 4 * 5 * 6 + 1 = 121 = 11^2, so 4 is in the sequence.

%e 6 * 7 * 8 + 1 = 337, which is a prime number, so 6 is not in the sequence.

%o (MAGMA) P<n> := PolynomialRing(Integers()); {k: k in Sort([ p : p in IntegralPoints(EllipticCurve(n^3 + 3*n^2 + 2*n + 1)) ])}; // _Mohamed Bouhamida_, Nov 29 2007

%o (PARI) isok(k) = issquare(k*(k+1)*(k+2)+1); \\ _Altug Alkan_, Dec 07 2015

%Y Cf. A258692.

%K fini,nonn,full

%O 1,2

%A _Roger Cuculière_, Aug 21 2006

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Last modified June 14 01:29 EDT 2021. Contains 345016 sequences. (Running on oeis4.)