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, Christopher Nov 29 '18 at 10:02

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Namaste, max_zorn, Brahadeesh, Ali Caglayan, Christopher
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ So I assume you mean the sum of 4 squares of rational numbers? $\endgroup$ – JDMan4444 Nov 29 '18 at 0:15
  • $\begingroup$ yes, thank you. $\endgroup$ – user2782067 Nov 29 '18 at 0:18
  • $\begingroup$ I don't understand, you can take $\frac{c}{d}$ ,$\frac{e}{f}$ , $\frac{g}{h}$ and $\frac{i}{j}$ square them and sum $\endgroup$ – Enigsis Nov 29 '18 at 0:22
  • $\begingroup$ @Enigsis It is the issue of showing, say, $1/7$ is equal to such a sum. $\endgroup$ – 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 $$ \left( \frac{w}{b} \right)^2 + \left( \frac{x}{b} \right)^2 + \left( \frac{y}{b} \right)^2 + \left( \frac{z}{b} \right)^2 = \frac{a}{b}$$


Not the answer you're looking for? Browse other questions tagged or ask your own question.