A116108 Squares that are equal to the sum of 3 consecutive cubes.

0, 9, 36, 41616

Offset: 1

Author

Zak Seidov, Apr 14 2007

Keywords

• fini
• nonn
• full

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

• J. W. S. Cassels, A Diophantine equation, Glasgow Math. J., 27 (1985), 11-18.
• Saburo Uchiyama, On a Diophantine equation, Proc. Japan Acad., Ser. A 55 (1979), 367-369.

Crossrefs

Cf. A027602, A218979.

Programs

Magma

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

Mathematica

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