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A133475 Integers n such that n^3 + n^2 - 9*n + 16 is a square. 0

%I #16 Sep 08 2022 08:45:31

%S -4,-3,-1,0,1,3,5,11,15,28,47,55,81,549,1799,8361

%N Integers n such that n^3 + n^2 - 9*n + 16 is a square.

%C The set of x values of integral points on the elliptic curve y^2 = x^3 + x^2 - 9*x + 16.

%e 0^3 + (-5)^2 + (-9) = 4^2, 1^3 + (-4)^2 + (-8) = 3^2, 3^3 + (-2)^2 + (-6) = 5^2

%t ok[x_] := Reduce[{y^2 == x^3 + x^2 - 9*x + 16, y >= 0}, y, Integers] =!= False; Select[Table[x, {x, -4, 10000}], ok ] (* _Jean-François Alcover_, Sep 07 2011 *)

%o (Magma) P<n> := PolynomialRing(Integers()); {x: x in Sort([ p[1] : p in IntegralPoints(EllipticCurve(n^3 + n^2 - 9*n + 16)) ])};

%o (Sage) EllipticCurve([0,1,0,-9,16]).integral_points()

%o (PARI) is(n)=issquare(n^3+n^2-9*n+16) \\ _Charles R Greathouse IV_, Sep 06 2016

%Y Cf. A117950, A132411, A132414, A002522, A028872.

%K sign,full,fini

%O 1,1

%A _Mohamed Bouhamida_, Nov 29 2007

%E Edited by _Max Alekseyev_, Nov 13 2009

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Last modified March 28 00:43 EDT 2024. Contains 371235 sequences. (Running on oeis4.)