The only solutions are , (0, 0), (1, 1), (5, 10), (6, 13), and (85, 645) (Guy 1994, p.147), corresponding to the nontrivial triangular square pyramidal numbers 1, 55, 91, 208335.
Numbers which are simultaneously tetrahedral and square pyramidal satisfy the Diophantine equation
Beukers (1988) has studied the problem of finding solutions via integral points on an elliptic curve and found that the only solution is the trivial .
Anglin, W.S. "The Square Pyramid Puzzle." Amer. Math. Monthly 97,120-124, 1990.
Anglin, W.S. The Queen of Mathematics: An Introduction to Number Theory. Dordrecht, Netherlands: Kluwer, 1995.
Baker, A. and Davenport, H. "The Equations and ." Quart J. Math. Ser. 2 20, 129-137, 1969.
Ball, W.W.R. and Coxeter, H.S.M. MathematicalRecreations and Essays, 13th ed. New York: Dover, p.59, 1987.
Beukers, F. "On Oranges and Integral Points on Certain Plane Cubic Curves."Nieuw Arch. Wisk. 6, 203-210, 1988.
Conway, J.H. and Guy, R.K. TheBook of Numbers. New York: Springer-Verlag, pp.47-50, 1996.
Dickson, L.E. History of the Theory of Numbers, Vol.2: Diophantine Analysis. New York: Dover, 2005.
Guy, R.K. "Figurate Numbers." §D3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp.147-150, 1994.
Kanagasabapathy, P. and Ponnudurai, T. "The Simultaneous Diophantine Equations and ." Quart. J. Math. Ser. 2 26, 275-278, 1975.
Ljunggren, W. "New Solution of a Problem Posed by E.Lucas." NordiskMat. Tidskrift 34, 65-72, 1952.
Lucas, É. Question 1180. Nouv. Ann. Math. Ser. 2 14, 336, 1875.
Lucas, É. Solution de Question 1180. Nouv. Ann. Math. Ser. 2 15,429-432, 1877.
Ma, D.G. "An Elementary Proof of the Solution to the Diophantine Equation ." Sichuan Daxue Xuebao 4, 107-116, 1985.
Moret-Blanc, M. Question 1180. Nouv. Ann. Math. Ser. 2 15, 46-48, 1876.
Ogilvy, C.S. and Anderson, J.T. Excursionsin Number Theory. New York: Dover, pp.77 and 152, 1988.
Sloane, N.J.A. Sequence A000330/M3844in "The On-Line Encyclopedia of Integer Sequences."
Watson, G.N. "The Problem of the Square Pyramid." Messenger. Math. 48,1-22, 1918.
Wolf, T. "The Puzzle." http://home.tiscalinet.ch/t_wolf/tw/misc/squares.html.
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