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This Wikipedia article of Infinite Descent says:

We have $ 3 \mid a_1^2+b_1^2 \,$. This is only true if both $a_1$ and $b_1$ are divisible by $3$.

But how can this be proved?

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    Do you know modular arithmetic? The only squares modulo three are zero and one, and so the sums of two squares can only be 0 if both of the squares are congruent to 0.2012-08-26

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