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How do you find a number that is the sum of two squares and whose square is also the sum of two squares?

1 Answer

Jul 31, 2017

If you take any two numbers, square them and add the squares, then the resulting number will satisfy your criteria.

Explanation:

Choose two numbers $a$ $a$ and $b$ $b$ .

Then:

${(a^{2} - b^{2})}^{2} + {(2 a b)}^{2} = (a^{4} - 2 a^{2} b^{2} + b^{4}) + 4 a^{2} b^{2}$ $(a^2-b^2)^2 + (2ab)^2 = (a^4-2a^2b^2+b^4) + 4a^2b^2$

${(a^{2} - b^{2})}^{2} + {(2 a b)}^{2} = a^{4} + 2 a^{2} b^{2} + b^{4}$ $color(white)((a^2-b^2)^2 + (2ab)^2) = a^4+2a^2b^2+b^4$

${(a^{2} - b^{2})}^{2} + {(2 a b)}^{2} = {(a^{2} + b^{2})}^{2}$ $color(white)((a^2-b^2)^2 + (2ab)^2) = (a^2+b^2)^2$

That is ${(a^{2} + b^{2})}^{2}$ $(a^2+b^2)^2$ is the sum of two squares too.

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