Square Pyramidal Number

A Figurate Number of the form

$$P_n={\textstyle{1\over 6}} n(n+1)(2n+1),$$ (1)

corresponding to a configuration of points which form a Square Pyramid, is called a square pyramidal number (or sometimes, simply a Pyramidal Number). The first few are 1, 5, 14, 30, 55, 91, 140, 204, ... (Sloane's A000330). They are sums of consecutive pairs of Tetrahedral Numbers and satisfy

$$P_n={\textstyle{1\over 3}}(2n+1)T_n,$$ (2)

where $T_n$ is the $n$th Triangular Number.

The only numbers which are simultaneously Square and pyramidal (the Cannonball Problem) are $P_1=1$ and $P_{24}=4900$, corresponding to $S_1=1$ and $S_{70}=4900$ (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

$$y^2={\textstyle{1\over 6}} x(x+1)(2x+1)$$ (3)

(Guy 1994, p. 147). However, an elementary proof has also been given by a number of authors.

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

$$3(2y+1)^2=8x^3+12x^2+4x+3.$$ (4)

The only solutions are $x=-1$, 0, 1, 5, 6, and 85 (Guy 1994, p. 147). Beukers (1988) has studied the problem of finding numbers which are simultaneously Tetrahedral and square pyramidal via Integer points on an Elliptic Curve. He finds that the only solution is the trivial ${\it Te}_1=P_1=1$.

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.