Diophantine Quadratic Equation in Three Variables

Solution

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For n = 1, we easily find that (5, 5, 3) is a solution. Given a little more time, we could find the solution (1, 3, 7) used in the book. Anyway, the induction tells us that, for any odd n > 0, the equation has a solution.

Obviously, for n = 2, finding a solution is a more difficult task. The equation is

a² + b² + c² = 59² = 3481.

This is the place where I would allow to use a calculator. One can spend pretty much time on trying to get past this hurdle. This is why I do not like this problem very much. Here's one solution (14, 39, 42). This allows us to complete the induction.

References

1. T. Andreescu, D. Andrica, I. Cucurezeanu, An Introduction to Diophantine Equations, Birkhäuser, 2010, p. 38

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