Cannonball problem

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.^[1] 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.^[2]^[3]

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.^[4]

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.

References

^ David Darling. "Cannonball Problem". The Internet Encyclopedia of Science.
^ Ma, D. G. (1985). "An Elementary Proof of the Solutions to the Diophantine Equation $6y^{2}=x(x+1)(2x+1)$ ". Sichuan Daxue Xuebao. 4: 107–116.
^ Anglin, W. S. (1990). "The Square Pyramid Puzzle". American Mathematical Monthly. 97 (2): 120–124. doi:10.2307/2323911. JSTOR 2323911.
^ "week95". Math.ucr.edu. 1996-11-26. Retrieved 2012-01-04.
^ Sloane, N. J. A. (ed.). "Sequence A039596 (Numbers that are simultaneously triangular and square pyramidal)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ ^a ^b Weisstein, Eric W. "Square Pyramidal Number". MathWorld.

External links

Weisstein, Eric W. "Cannonball Problem". MathWorld.

[1] David Darling. "Cannonball Problem". The Internet Encyclopedia of Science.

[2] Ma, D. G. (1985). "An Elementary Proof of the Solutions to the Diophantine Equation $6y^{2}=x(x+1)(2x+1)$ ". Sichuan Daxue Xuebao. 4: 107–116.

[3] Anglin, W. S. (1990). "The Square Pyramid Puzzle". American Mathematical Monthly. 97 (2): 120–124. doi:10.2307/2323911. JSTOR 2323911.

[4] "week95". Math.ucr.edu. 1996-11-26. Retrieved 2012-01-04.

[5] 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-6] Weisstein, Eric W. "Square Pyramidal Number". MathWorld.

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