The equation \( y^2 = x^3 - x + 5 \) has no integer solutions because when analyzed modulo 3, it simplifies to \( y^2 \equiv 2 \pmod{3} \). According to Fermat's Little Theorem, \( x^3 \equiv x \pmod{3} \), leading to the conclusion that \( y^2 \equiv 2 \). Since 2 is a quadratic non-residue modulo 3, there cannot be any integer \( y \) that satisfies the equation. Therefore, the equation has no integer solutions.