Find all the natural solutions to this diophantine equation

Question

Find all the natural solutions to this diophantine equation $968m =n^2-54257$

Answer 1

As $54257\equiv49\pmod {968}$ and $\displaystyle 968m=n^2-54257\implies n^2\equiv54257 \pmod {968}$

$\displaystyle\implies n^2\equiv49=7^2\pmod {968}$

As $\displaystyle968=2^3\cdot11^2$

we need $\displaystyle(i)n^2\equiv49\pmod{2^3}\equiv1$ (See Prove that $x^{2} \equiv 1 \pmod{2^k}$ has exactly four incongruent solutions)

and $\displaystyle(i)n^2\equiv49\pmod{11^2}$
Use Solutions of the congruence $x^2 \equiv 1 \pmod{m}$ or solution set for congruence $x^2 \equiv 1 \mod m$