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) |
See also Tetrahedral Number
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 Eric W. Weisstein