回复：Your Ex,5should be 5^2m,3 ...3^2n...,We get |9781-Z^2|=1,

My current understanding is that there are infinite integral solutions of x^2 - 9781*y^2 = 1. Euler(?)  developed some standard method to caculate the least nontrivial integral solution.

I really appreciate the discussion with you and I was enlightened by this discussion a lot. My initial understanding of these diophatine equtions is zero, now I picked up a little. I like this kind of learning process.