A Figurate Number of the form

(1) |

(2) |

The only numbers which are simultaneously Square and pyramidal (the Cannonball Problem) are
and , corresponding to and (Dickson 1952, p. 25; Ball and Coxeter 1987, p. 59;
Ogilvy 1988), as conjectured by Lucas (1875, 1876) and proved by Watson (1918). The proof is far from elementary, and
is equivalent to solving the Diophantine Equation

(3) |

Numbers which are simultaneously Triangular and square pyramidal satisfy
the Diophantine Equation

(4) |

**References**

Ball, W. W. R. and Coxeter, H. S. M. *Mathematical Recreations 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. *The Book 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: Chelsea, 1952.

Guy, R. K. ``Figurate Numbers.'' §D3 in
*Unsolved Problems in Number Theory, 2nd ed.* New York: Springer-Verlag, pp. 147-150, 1994.

Lucas, É. Question 1180. *Nouvelles Ann. Math. Ser. 2* **14**, 336, 1875.

Lucas, É. Solution de Question 1180. *Nouvelles Ann. Math. Ser. 2* **15**, 429-432, 1876.

Ogilvy, C. S. and Anderson, J. T. *Excursions in Number Theory.* New York: Dover, pp. 77 and 152, 1988.

Sloane, N. J. A. Sequence
A000330/M3844
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

Watson, G. N. ``The Problem of the Square Pyramid.'' *Messenger. Math.* **48**, 1-22, 1918.

© 1996-9

1999-05-26