$x^2 + y^2$

Number Theory Level 2

Charles and Henry are discussing:

Charles: 158 cannot be written as a sum of two perfect squares.

Henry: No, you are wrong. It can be written as a sum of two perfect squares.

Who is correct?

Both of them Neither of them Henry Charles

1 solution

Anirudh Sreekumar
Oct 18, 2017

A perfect square can be $0,1$ or $4$ $\pmod 8$

So, the sum of $2$ perfect squares will be $0,1,2,4,5$ $\pmod8$

$158 \pmod 8=6 \pmod 8$

$\implies 158$ cannot be the sum of $2$ squares