modular arithmetic proof question

Question

I have to show that there are no integers x and y satisfying the equation 7$x^2-15y^2=1$.

I don't know from where to start any hints. Thanks

Answer 1

Observe that

$$1=7x^2-15y^2=2x^2\pmod 5\implies x^2=\frac12=3\pmod 5$$

which is impossible as $\;3\;$ isn't a quadratic residue modulo $\;5\;$.

Answer 2

That $3$ isn't a quadratic residue mod $5$ follows from Euler's criterion: in this case, $3^{\frac{5-1}2}=9\cong -1\pmod5$.

I.e. the Legendre symbol $\left(\frac53\right) =-1$.