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

  • $\begingroup$ Simplify the equation by considering it mod $n$ for various choices of $n$ and see if that helps. For example, I would consider the equation mod $7$ and mod $15$ since they both simplify the equation considerably. $\endgroup$ – Malcolm Apr 7 '18 at 16:33

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 $\;5\;$.


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 $\left(\frac53\right) =-1$.


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