A136276 Consider pairs of nonnegative integers (m,k) such that 2^2 + 4^2 + 6^2 + ... + (2m)^2 = k(k+1); sequence gives k values.

0, 4, 7, 84

Offset: 1

Author

Ken Knowlton (www.KnowltonMosaics.com), Mar 29 2008

Keywords

• nonn
• fini
• full

Comments

The problem arises when trying to build a square pyramid out of dominoes. The solution (m,k) = (3,7) for example corresponds to building a pyramid with layers of sizes 2 X 2, 4 X 4 and 6 X 6 from one set of double-6 dominoes.

The three nonzero solutions use one double-3 set, one double-6 set and one double-83 set. (The sequence 3,6,83 is too short to warrant a separate entry.)

The problem is equivalent to finding integers (m,k) such that 2m(m+1)(m+2)/3 = k*(k+1). This is a nonsingular cubic, so by Siegel's theorem, there are only finitely many solutions. - N. J. A. Sloane, May 25 2008. See also the articles by Stroeker and Tzanakis and Stroeker and de Weger.

Examples

			The known solutions are (m,k) = (0,0), (2,4), (3,7) and (17,84). There are no other solutions.
		

Links

• John Cannon, Using MAGMA to prove there are no other solutions
• J. H. Silverman and J. Tate, Rational Points on Elliptic Curves, Springer, 1992,
• R. J. Stroeker and B. M. M. de Weger, Solving elliptic Diophantine equations: the general cubic case, Acta Arith. 87 (1999), 339-365.
• R. J. Stroeker and N. Tzanakis, Solving elliptic Diophantine equations by estimating linear forms in elliptic logarithms, Acta Arith. 67 (1994), 177-196.

Crossrefs

Cf. A039596, A053611, A053612.

Programs

Maple

Simple-minded Maple program from N. J. A. Sloane:
f1:=m-> 1+8*m*(m+1)*(2*m+1)/3;
for m from 0 to 10^8 do if issqr(f1(m)) then lprint( m, (-1+sqrt(f1(m)))/2); fi; od;

Extensions

Edited by N. J. A. Sloane, May 25 2008, Aug 17 2008

May 26 2008: John Cannon used MAGMA to show there are no further solutions (see link)