Prove that there are no rational numbers $x$ and $y$ such that $x^2+y^2=1002$

Asked Dec 01 '22 at 19:53

Active Dec 04 '22 at 09:50

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Prove that there are no rational numbers $x$ and $y$ such that $x^2+y^2=1002$.

Hello,

I am completely new to proofs and am having a hard time with this one. Can someone help me with it?

I figured we go with proof by contradiction and assume x and y are rational thus,

$$ x = \frac{p}{q} , \ y = \frac{s}{t}, \text{where} \ p,q,s,t \in \ Z \ \text{and} \ q , t\neq \ 0 ; $$

But where should I go from there? Should I substitute that to $x^2+y^2=1002$ to get

$$ \left(\frac{p}{q}\right)^2 + \left(\frac{s}{t}\right)^2 = 1002 $$

to then simplify? Any advice is appreciated.

Thanks

edited Dec 04 '22 at 09:50

Gabriel Soranzo

asked Dec 01 '22 at 19:53

Wsiz

2

See https://math.stackexchange.com/questions/1445058/which-rationals-can-be-written-as-the-sum-of-two-rational-squares – lhf Dec 01 '22 at 20:00
1

$1002$ is divisible by $3$ but it isn't divisible by $9$. – Jyrki Lahtonen Dec 01 '22 at 22:16

0 Answers0