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
(2)
The square pyramidal numbers are sums of consecutive pairs of tetrahedral
numbers and satisfy
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
(4)
(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).
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.
Anglin, W. S. "The Square Pyramid Puzzle." Amer. Math. Monthly97, 120-124, 1990.Anglin, W. S.
The
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R. K. The
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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,
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Equations
and ."
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Nouv. Ann. Math. Ser. 215, 429-432, 1877.Ma, D. G.
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Excursions
in Number Theory. New York: Dover, pp. 77 and 152, 1988.Sloane,
N. J. A. Sequence A000330/M3844
in "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.