Square Pyramidal Number

A figurate number of the form

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, ... (OEIS A000330). The generating function for square pyramidal numbers is

The square pyramidal numbers are sums of consecutive pairs of tetrahedral numbers and satisfy

where is the th triangular number.

The only numbers which are simultaneously square and square pyramidal (the cannonball problem) are and , corresponding to and (Ball and Coxeter 1987, p. 59; Ogilvy 1988; Dickson 2005, p. 25), as conjectured by Lucas (1875), partially proved by Moret-Blanc (1876) and Lucas (1877), and proved by Watson (1918). The problem requires solving the Diophantine equation

(Guy 1994, p. 147). Watson (1918) gave an almost elementary proof, disposing of most cases by elementary means, but resorting to the use of elliptic functions for one pesky case. Entirely elementary proofs have been given by Ma (1985) and Anglin (1990).

Numbers which are simultaneously triangular and square pyramidal satisfy the Diophantine equation

Completing the square gives

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 .