Does the Equation \( y^2 = x^3 - x + 5 \) Have Integer Solutions?

[Poirot1]

Us the fact that 2 is not a quadratic residue of 3 to show that there are no integer solutions to $y^2=x^3-x+5$.

 

[Nono713]

Re: show equation has no solution

Take the equation modulo $3$, and you get:
$$y^2 \equiv x^3 - x + 5 \equiv x^3 - x + 2 \pmod{3}$$
Now FLT tells us that $x^3 \equiv x \pmod{3}$, so we get:
$$y^2 \equiv x - x + 2 \equiv 2 \pmod{3}$$
Can you finish?

 

[Poirot1]

Re: show equation has no solution

no such y exist because 2 is a quadractic non-residue mod 3.