Every four consecutive integers contains one which cannot be written as sum of two squares

Question

Every four consecutive integers contains one which cannot be written as sum of two squares.

Could anyone advise me how to prove the statement? Do I use Jacobi's two squares theorem? Hints will suffice, thank you.

Answer 1

Hint:

• There is a number of the form of $4k+3$.

• You might like to explore $x^2 \pmod{4}$.

Answer 2

You may consider the following as a hint:

Squares are of the form $4k$ or $4k+1$ "easy to verify $(2k)^2$,$(2k+1)^2$". Now, what are the possible forms of the summation of these numbers?

Answer 3

One further hint: Consider that Pythagorean triples $x^2+y^2=z^2$ can be found as $x=m^2-n^2,\space y=2mn,\space z=m^2+n^2$.