Hello,
the task is to solve (find all integer solutions) following quadratic Diophantine equation:
$\displaystyle x^2-4y^2=65$

I think I need to use modular arithmetic for obtaining solutions. Maybe I should start investigating squares of integers $\displaystyle mod \ x$, $\displaystyle x = 1,2,...$

Also I peeled answer from WolframAlpha, but that doesn't give me any hint how to get it myself. WolframAlpha gave the following solutions:
$\displaystyle x = \pm 33, \ y = \pm 16$
$\displaystyle x = \pm 9, \ y = \pm 2$

So, any help is appreciated. Thanks!

2. I recommend factoring the left hand side:

$\displaystyle (x+2y)(x-2y)=65$

Now the terms on the left must be factors of 65. There are several choices: 1 and 65, 65 and 1, 5 and 13, 13 and 5 (and all of the appropriate negative pairs). For example,

$\displaystyle x+2y=1$
$\displaystyle x-2y=65$

Solving the system gives $\displaystyle x=33,y=-16$. Try it for all of the other pairs and you will get your entire solution set.

Good luck.