The sum of the first $n$ squares is a square: a system of two Pell-type-equations

number-theory diophantine-equations square-numbers pell-type-equations sums-of-squares

Solution 1:

This problem is sometimes called cannonball problem or square pyramid problem.

Maybe you may have a look at MathWorld or Wikipedia and some references therein.

Elementary solution is given in W. S. Anglin: The Square Pyramid Puzzle, The American Mathematical Monthly, Vol. 97, No. 2 (Feb., 1990), pp. 120-124.

Further useful references:

  • Henri Cohen: Number Theory: Tools and diophantine equations, Section 6.8.2

  • S. C. Althoen and C. B. Lacampagne: Tetrahedral Numbers as Sums of Square Numbers, Mathematics Magazine, Vol. 64, No. 2 (Apr., 1991), pp. 104-108

Online resources:

  • Wolfram MathWorld
  • Wikipedia

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