Cannonball problem

A square pyramid of cannonballs in a square frame

In the mathematics of figurate numbers, the cannonball problem asks which numbers are both square and square pyramidal. The problem can be stated as: given a square arrangement of cannonballs, for what size squares can these cannonballs also be arranged into a square pyramid. Equivalently, which squares can be represented as the sum of consecutive squares, starting from 1?

Formulation as a Diophantine equation

When cannonballs are stacked within a square frame, the number of balls is a square pyramidal number; Thomas Harriot gave a formula for this number around 1587, answering a question posed to him by Sir Walter Raleigh on their expedition to America. Édouard Lucas formulated the cannonball problem as a Diophantine equation

$\sum _{n=1}^{N}n^{2}=M^{2}$

or

${\frac {1}{6}}N(N+1)(2N+1)={\frac {2N^{3}+3N^{2}+N}{6}}=M^{2}.$

Solution

Lucas conjectured that the only solutions are N = 1, M = 1, and N = 24, M = 70, using either 1 or 4900 cannon balls. It was not until 1918 that G. N. Watson found a proof for this fact, using elliptic functions. More recently, elementary proofs have been published.

Applications

The solution N = 24, M = 70. can be used for constructing the Leech Lattice. The result has relevance to the bosonic string theory in 26 dimensions.

Although it is possible to tile a geometric square with unequal squares, it is not possible to do so with a solution to the cannonball problem. The squares with side lengths from 1 to 24 have areas equal to the square with side length 70, but they cannot be arranged to tile it.

Related problems

The only numbers that are simultaneously triangular and square pyramidal, are 1, 55, 91, and 208335..

There are no numbers (other than the trivial solution 1) that are both tetrahedral and square pyramidal.

See also

• Square triangular number, the numbers that are simultaneously square and triangular
• Sixth power, the numbers that are simultaneously square and cubical