# Express rational number as the sum of 4 squares [closed]

I need to prove that any rational number a/b can be expressed as the sum of 4 squares of rational numbers

I have tried something similar to the proof for integers, but can't get it to work. I have tried an algebraic proof, but I keep running into the issie of potential irrational denominators

## closed as off-topic by Namaste, max_zorn, Brahadeesh, Ali Caglayan, ChristopherNov 29 '18 at 10:02

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• So I assume you mean the sum of 4 squares of rational numbers? – JDMan4444 Nov 29 '18 at 0:15
• yes, thank you. – user2782067 Nov 29 '18 at 0:18
• I don't understand, you can take $\frac{c}{d}$ ,$\frac{e}{f}$ , $\frac{g}{h}$ and $\frac{i}{j}$ square them and sum – Enigsis Nov 29 '18 at 0:22
• @Enigsis It is the issue of showing, say, $1/7$ is equal to such a sum. – JDMan4444 Nov 29 '18 at 0:23

You are taking $$\frac{a}{b}$$ with integers $$a,b > 0.$$ Thus $$ab$$ is a positive integer.
Let $$w^2 + x^2 + y^2 + z^2 = ab$$
Then $$\left( \frac{w}{b} \right)^2 + \left( \frac{x}{b} \right)^2 + \left( \frac{y}{b} \right)^2 + \left( \frac{z}{b} \right)^2 = \frac{a}{b}$$