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?
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
and conjectured that the only solutions are N = 1, M = 1, and N = 24, M = 70. 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. 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.
- Square triangular number, the numbers that are simultaneously square and triangular
- Sixth power, the numbers that are simultaneously square and cubical
- David Darling. "Cannonball Problem". The Internet Encyclopedia of Science.
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- Anglin, W. S. (1990). "The Square Pyramid Puzzle". American Mathematical Monthly. 97 (2): 120–124. doi:10.2307/2323911. JSTOR 2323911.
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- Sloane, N.J.A. (ed.). "Sequence A039596 (Numbers that are simultaneously triangular and square pyramidal)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- Weisstein, Eric W. "Square Pyramidal Number". MathWorld.
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