I'm struggling with proving there are no solutions or giving the parametric solution for Diophantine equations: $$ \text{ a) } 15x^2 -7y^2 =9$$ $$ \text{ b) } x^2 + y^2 = 5z+3$$
For a) I've tried using modulo 4 $\text{ LHS }15x^2 -7y^2$ is congruent to $0$ or $1$ (mod $4$) $ \text{ RHS } 9$ is congruent to $3$ (mod $4$) so this implies there are no integer solutions.
For b) I think there are solutions from just plugging in numbers but I cannot find the parametric solutions. LHS: $x^2 + y^2$ is congruent to $0,1, 4$ (mod $5$) and RHS: $5z + 3$ is congruent to $3$ (mod $5$)
From here I'm lost with what to do. Can someone show me and explain what is going on. Thanks
mod 4 doesn't work for (a). But mod 5 does. It is easy to check that $-7y^2\ne4\bmod5$. So there are no integer solutions.
For (b) we need $x^2+y^2=3\bmod5$. Squares are $0,\pm1$ mod 5. So we need both $x^2$ and $y^2$ to be -1 mod 5, and hence both $x$ and $y$ to be $\pm2$ mod 5. It 0 easy to get $z$. There are 4 cases. For example: $x=5a+2,y=5b+2$. Then $x^2+y^2=25a^2+20a+4+25b^2+20b+4$ and so $$z=5a^2+5a+5b^2+4b+1$$ and similarly for the other cases.