2026/049) Does there exist integer solutions to $x^2+y^2=2022$

Before we look for solutions let us quickly check based on modular arithmetic.

 Looking at mod 3 we have 2022 is 2 mod 4. 

Solution may exist.

Let us check based on mod 3.

We have 

$2022 \equiv 0 \pmod 3$

But $2022 \equiv 6 \pmod 9$

So 2022 is not sum of 2 perfects squares

Let us prove the basis of the same

x is of the form $3a$ or $3a+1 or $3a+2$

If $x \equiv 0 \pmod 3$ then $x^2 \equiv 0 \pmod 3$

If $x \equiv 1 \pmod 3$ then $x^2 \equiv 1 \pmod 3$

If $x \equiv 2 \pmod 3$ then $x^2 \equiv 1 \pmod 3$

Similarly for y

Now looking at above we have $x^2+y^2 \equiv 0 \pmod 3 $ iff $x \equiv 0 \pmod 3$ and  $y \equiv 0 \pmod 3$.

So x = 3a and y = 3b for some a and b

or $x^2+y^2 = 9(a^2+b^2)$ or divisible by 9

But 2022 is not divisible by 9

Hence no solution