Are the solutions of ?
Hi all, here is a little conjecture (I do have my proof but it could be interesting to see others' proofs), let's see how you do.
Say with and primes ( is a semiprime, thus).
Prove that there exists only two distinct solution pairs such that with , .
Good luck !
Hello. I noticed that you didn't actually receive a (formal) solution, so I thought that I would give mine.
I will rearrange the equation in question:
And once more:
We know for sure that the terms and are divisors of . Denoting as arbitrary divisors of , it suffices to solve all of the systems of the form ; .
The systems have integer solutions precisely when and . In particular, and , respectively. (Note that it may be necessary to reverse the roles of to adjust the sign of .)
Other combinations do not work for any semiprime or lead to redundancy when one of the factors is 2. For example, gives . We can see that is always nonpositive, etc...
I think that this does it, for the most part.
Thanks for the problem.
Interesting, indeed. My proof uses the fact that this equation can be considered as being the discriminant of the quadratic equation , where . Therefore I can easily prove that there are only two solutions which are ... etc ...
Good idea though, wouldn't have seen it like that