A121234 -id:A121234 - cp's OEIS frontend

A258692 Integers n such that n(n + 2)(n + 4) + 1 is a perfect square.

-4, -3, -2, 0, 1, 2, 8, 10, 18, 112, 1272

Offset: 1

Morris Neene, Jun 12 2015

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.

			1 * 3 * 5 + 1 = 16 = 4^2, so 4 is in the sequence.
2 * 4 * 6 + 1 = 49 = 7^2, so 2 is in the sequence.
3 * 5 * 7 + 1 = 106 = 2 * 53, so 3 is not in the sequence.

Cf. A121234.

P := PolynomialRing(Integers()); {x: x in Sort([ p[1] : p in IntegralPoints(EllipticCurve(n^3 + 6*n^2 + 8*n + 1)) ])};

Select[Range[-10, 100], IntegerQ[Sqrt[#(# + 2)(# + 4) + 1]] &] (* Alonso del Arte, Jun 12 2015 *)

[i[0] for i in EllipticCurve([0, 6, 0, 8, 1]).integral_points()] # Seiichi Manyama, Aug 26 2019

A334189 Positive solutions m of the Diophantine equation x * (x+1) * (x+2) = y * (y+1) * (y+2) * (y+3) = m.

Original entry on oeis.org

24, 120, 175560

Offset: 1

Bernard Schott, Apr 18 2020

Boyd and Kisilevsky in 1972 proved that there exist only 3 solutions (x,y) = (2,1), (4,2), (55,19) to the Diophantine equation x * (x+1) * (x+2) = y * (y+1) * (y+2) * (y+3) [see the reference and a proof in the link].

A similar result: in 1963, L. J. Mordell proved that (x,y) = (2,1), (14,5) are the only 2 solutions to the Diophantine equation x * (x+1) = y * (y+1) * (y+2) with 2*3 = 1*2*3 = 6 and 14*15 = 5*6*7 = 210.

			24 = 2*3*4 = 1*2*3*4;
120 = 4*5*6 = 2*3*4*5;
175560 = 55*56*57 = 19*20*21*22.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Revised edition), Penguin Books, 1997, entry 175560, p. 175.

David. W. Boyd and Hershy Kisilevsky, The diophantine equation u(u+1)(u+2)(u+3) = v(v + 1)(v + 2), Pacific J. Math. 40 (1972), 23-32.

Cf. A121234.

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A258692 Integers n such that n(n + 2)(n + 4) + 1 is a perfect square.

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A334189 Positive solutions m of the Diophantine equation x * (x+1) * (x+2) = y * (y+1) * (y+2) * (y+3) = m.

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cp's OEIS Frontend

A258692 Integers n such that n*(n + 2)*(n + 4) + 1 is a perfect square.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

SageMath

A334189 Positive solutions m of the Diophantine equation x * (x+1) * (x+2) = y * (y+1) * (y+2) * (y+3) = m.

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A258692 Integers n such that n(n + 2)(n + 4) + 1 is a perfect square.