

A116108


Squares that are equal to the sum of 3 consecutive cubes.


12




OFFSET

1,2


COMMENTS

m^3+(m+1)^3+(m+2)^3=3(1+m)*(3+2*m+m^2). Corresponding values of m are 1,0,1,23.
The equation s^2 = 3c^3 + 6c can be transformed using the substitution X = 3c, Y = 3s into Y^2 = X^3 + 18X, a form of the Weierstrass equation of an elliptic curve: Y^2 = X^3 + aX^2 + bX + c, with a = c = 0. We can now use the Sage program to show that there are no other integer solutions.  Jaap Spies, May 27 2007
Confirmed by MAGMA  see code below.  Warut Roonguthai, May 28 2007
That there are no other integer solutions is a theorem of Uchiyama, rediscovered by Cassels. For n consecutive cubes summing to a square, see A218979.  Jonathan Sondow, Apr 03 2014


LINKS

Table of n, a(n) for n=1..4.
J. W. S. Cassels, A Diophantine equation, Glasgow Math. J., 27 (1985), 1118.
Saburo Uchiyama, On a Diophantine equation, Proc. Japan Acad., Ser. A 55 (1979), 367369.


MATHEMATICA

Select[Total/@Partition[Range[2, 200]^3, 3, 1], IntegerQ[Sqrt[#]]&] (* Harvey P. Dale, Aug 08 2015 *)


PROG

(MAGMA) IntegralPoints(EllipticCurve([18, 0]));


CROSSREFS

Cf. A027602, A218979.
Sequence in context: A203764 A053949 A071134 * A232257 A091961 A103758
Adjacent sequences: A116105 A116106 A116107 * A116109 A116110 A116111


KEYWORD

fini,nonn,full


AUTHOR

Zak Seidov, Apr 14 2007


STATUS

approved



