Prove for the following set of Diophantine equations.

Question

Prove that there are no integers $q$, $m$, $n$ and $p$ for the following Diophantine equations:-

1. $$ 7m^2 = 1 + q^2$$
2. $$p^2 - 1 = 7n^2$$

Answer 1

$(1)$ $7m^2=q^2+1$

Write $q=7a+r, r \in [0, 6]$. Then, $7$ is to divide into $r^2+1$. But none of $1, 2, 5, 10, 17, 26, 37$ is divisible by $7$.

$(2) 7n^2=p^2-1$

This statement is wrong because $p=8$ and $n=3$ satisfy the Diophantine equation.