# Diophantine Quadratic Equation in Three Variables

Prove that for all positive integers n, the equation

x² + y² + z² = 59^{ n}

is solvable in positive integers.

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Copyright © 1996-2018 Alexander BogomolnyProve that for all positive integers n, the equation

x² + y² + z² = 59^{ n}

is solvable in positive integers.

First of all observe that if (a, b, c) is a solution, with

(59a)² + (59b)² + (59c)² = 59²(a² + b² + c²) = 59²59^{k} = 59^{k + 2}.

From here, if we have a solution for n = 1 we can derive the solvability of the equation for all n odd. Similarly, having a solution for

For n = 1, we easily find that (5, 5, 3) is a solution. Given a little more time, we could find the 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

### References

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

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Copyright © 1996-2018 Alexander Bogomolny