OFFSET
1,2
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. (End)
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.
EXAMPLE
The known solutions are (m,k) = (0,0), (2,4), (3,7) and (17,84). There are no other solutions.
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;
CROSSREFS
KEYWORD
nonn,fini,full
AUTHOR
Ken Knowlton (www.KnowltonMosaics.com), Mar 29 2008
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)
STATUS
approved