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.
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Formulation as a Diophantine equation
editWhen 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
or
Solution
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Lucas conjectured that the only solutions are (N,M) = (0,0), (1,1), and (24,70), using either 0, 1, or 4900 cannonballs. 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
editThe 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.
Related problems
editA triangular-pyramid version of the cannonball problem, which is to yield a perfect square from the Nth Tetrahedral number, would have N = 48. That means that the (24 × 2 = ) 48th tetrahedral number equals to (702 × 22 = 1402 = ) 19600. This is comparable with the 24th square pyramid having a total of 702 cannonballs.[5]
Similarly, a pentagonal-pyramid version of the cannonball problem to produce a perfect square, would have N = 8, yielding a total of (14 × 14 = ) 196 cannonballs.[6]
The only numbers that are simultaneously triangular and square pyramidal are 1, 55, 91, and 208335.[7][8]
There are no numbers (other than the trivial solution 1) that are both tetrahedral and square pyramidal.[8]
See also
edit- Square triangular number, the numbers that are simultaneously square and triangular
- Close-packing of equal spheres
References
edit- ^ Darling, David. "Cannonball Problem". The Internet Encyclopedia of Science.
- ^ Ma, De Gang (1984). "An Elementary Proof of the Solutions to the Diophantine Equation ". Chinese Science Bulletin. 29 (21): 1343–1343. doi:10.1360/csb1984-29-21-1343.
- ^ 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 A000292 (Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A002411 (Pentagonal pyramidal numbers: a(n) = n^2*(n+1)/2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ 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.
