## Problem 3

Find all integral solutions to $a^{2}+b^{2}+c^{2}=a^{2} b^{2}$.


## Solution

Answer: $0,0,0$.

Squares must be 0 or $1 \bmod 4$. Since the rhs is a square, each of the squares on the 1 hs must be $0 \bmod 4$. So a, $\mathrm{b}, \mathrm{c}$ are even. Put $\mathrm{a}=2 \mathrm{a}_{1}, \mathrm{~b}=2 \mathrm{~b}_{1}, \mathrm{c}=2 \mathrm{c}_{1}$. Then $\mathrm{a}_{1}{ }^{2}+\mathrm{b}_{1}{ }^{2}+\mathrm{c}_{1}{ }^{2}=$ square. Repeating, we find that $\mathrm{a}, \mathrm{b}, \mathrm{c}$ must each be divisible by an arbitrarily large power of 2. So they must all be zero.

