Find a way to stack a square of cannonballs laid out on the ground into a square pyramid (i.e., find a square number which is also square pyramidal). This corresponds to solving the Diophantine equation
 |
for some pyramid height 
.
The only solutions are 
and 
(Ball and Coxeter 1987, Dickson 2005), as conjectured
by Lucas (1875), partially proved by Moret-Blanc (1876) and Lucas (1877), and proved
by Watson (1918). Watson's proof was almost elementary, 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).
and 
." Quart J. Math. Ser. 2 20, 129-137,
1969.Ball, W. W. R. and Coxeter, H. S. M. 
and 
." Quart. J. Math. Ser. 2 26,
275-278, 1975.Ljunggren, W. "New Solution of a Problem Posed by
E. Lucas." Nordisk Mat. Tidskrift 34, 65-72, 1952.Lucas,
É. Question 1180. Nouv. Ann. Math. Ser. 2 14, 336, 1875.Lucas,
É. Solution de Question 1180. Nouv. Ann. Math. Ser. 2 15, 429-432,
1877.Ma, D. G. "An Elementary Proof of the Solutions to the
Diophantine Equation 
."
Sichuan Daxue Xuebao, No. 4, 107-116, 1985.Moret-Blanc,
M. Question 1180. Nouv. Ann. Math. Ser. 2 15, 46-48, 1876.Ogilvy,
C. S. and Anderson, J. T.