1. ## Another proof idea

Opalg, to continue the argument from your first post, that $\displaystyle pa^2-qb^2=1,pc^2-qd^2=-1$ has no solution for $\displaystyle p,q$ coprime and squarefree...

Given $\displaystyle a,b,c,d$, the solution is $\displaystyle p=\frac{b^2+d^2}D,q=\frac{a^2+c^2}D, D=(ad)^2-(bc)^2$ Can it be shown that $\displaystyle D\not|gcd(b^2+d^2,a^2+c^2)$, and therefore no integer solutions to $\displaystyle p,q$ exist unless $\displaystyle D=1$? I think you will be able to see that if $\displaystyle D=1$ then $\displaystyle p$ or $\displaystyle q$ must be $\displaystyle 1$, proving the theorem. I have gone only far enough to show $\displaystyle a,d$ must be odd and $\displaystyle b,c$ must be even, making $\displaystyle D\equiv1\bmod4$.

2. ## A proof, at last

Attached is a write-up of a final proof (it's three pages long). This problem was fascinatingly complicated. I knew virtually nothing about the Pell equation and continued fractions, so I have had the pleasant opportunity these last few days to understand how they work. Number theory really is enchanting the way a simple question can lead to an arbitrarily complex answer.

havaliza: Where the hell did this question come from? A book? A class? Your own wanderings?

After reading and understanding the attached proof, I wonder if anyone wants to look at a generalization involving x(y+n),y(x+n)...

3. I'll save it on my computer and read it in my next free time.

Number theory really is enchanting the way a simple question can lead to an arbitrarily complex answer.
What the hell was Fermat thinking when he wrote down $\displaystyle x^n + y^n = z^n$ attached with his very concise proof ?

4. You should probably want to open a thread to Pre-prints and other original work sub-forum. This is a quite interesting problem (and eventually) with its proof.

5. Originally Posted by Media_Man
Attached is a write-up of a final proof (it's three pages long). This problem was fascinatingly complicated. I knew virtually nothing about the Pell equation and continued fractions, so I have had the pleasant opportunity these last few days to understand how they work. Number theory really is enchanting the way a simple question can lead to an arbitrarily complex answer.

havaliza: Where the hell did this question come from? A book? A class? Your own wanderings?

After reading and understanding the attached proof, I wonder if anyone wants to look at a generalization involving x(y+n),y(x+n)...