Show that there is no point P = (x, y) with integer coordinates of the parabola

Question

Show that there is no point $P = (x, y)$ with integer coordinates of the parabola $y=\frac{1}{7}x^2-\frac{3}{7}$

It is part of number theory and congruences but I really do not know how to solve the above task? Can someone help me?

Answer 1

Multiplying both sides by $7$, you are asked to show that the equation $$7y=x^{2}-3$$ has no solutions over $\mathbb{Z}$. Assume by contradiction that $(x_{0},y_{0})\in\mathbb{Z}^{2}$ is a solution. Reducing the equation $7y_{0}=x_{0}^{2}-3$ modulo $7$ then gives $$x_{0}^{2}\equiv 3\ \text{mod}\ 7.$$ Hence $3$ would be a square modulo $7$, which is a contradiction. (Since $0,1,2$ and $4$ are the only squares modulo $7$).