Use congruence mod 4 to show that x is even in any integer s

Use congruence mod 4 to show that x is even in any integer solution of y³=x²+1. From now on assume that (x,y) is such a solution.

We prove by contradiction

Assume, x,y exist which solve

y^3=x^2+1 and x is odd

HEnce, x^2=1 modulo 4

HEnce, y^3=1+1=2 modulo 4

So, y^3 is even

HEnce, y is even

y=2m for some integer

y^3=8m^3=0 modulo m

HEnce a contradiction

Hence, x is even in any integer solution to y^3=x^2+1