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Square Numbers

Permanent link to this puzzle: mscroggs.co.uk/puzzles/48
Towards the end of his life, Lewis Carroll recorded in his diary that he had discovered that double the sum of two square numbers could always be written as the sum of two square numbers. For example
$$2(3^2 +4^2 )=1^2 +7^2$$ $$2(5^2 +8^2 )=3^2 +13^2$$
Prove that this can be done for any two square numbers.

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If you enjoyed this puzzle, check out Sunday Afternoon Maths LIX,
puzzles about 3d shapes, or a random puzzle.
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