I just need to know if this holds :
For any coprime with .
I know this is true but what about this :
For any coprime with .
This would seem to work intuitively but I'm having trouble proving or disproving it (all my tests seem to fail for some reason but not because of the math).
Thank you all
Therefore we require or ; this last part is trivial though.
When we have so the above line can't have a solution.
If , so we want to solve which we can do. Note in this case can be even too. This is because if the exponent doesn't matter and if then . I'd consider to be a trivial case.
So in summary there is only one case where you're statement holds, namely .
Edit: This post is also wrong. See below.
Haha looks like my question spilled a lot of virtual ink.
Thank you for your replies, with your help I managed to set up something that I can prove. Here's a nice theorem (don't know if it already exists ?) :
Let , , , ...
Assume are relative numbers with the property that for all . Then the following holds :
I will post the proof in some time. I do have it but it needs ... editing. It's quite simple, based on substituting zeroes modulo various .
Coming back to the original question, it is indeed a matter of greatest common divisors. For instance, I can hopefully prove the original statement using a variation of my general proof:
Due to Euler's Theorem this may only work if . And again due to Euler's Theorem, we can write the following congruence from the statement :
This, again, can only work if . Reapplying Euler's Theorem again the following congruence is obtained :
With does hold for any . Thus the original statement holds iff and .
Is this a valid proof ?
This whole thread has relevance for what I'd consider a fun/beautiful problem over at Project Euler: (spoiler box contains link)
I don't like giving hints that are too substantial or giving away answers, but it's such a nice tie-in I couldn't resist. Anyway, knowing that this thread is relevant doesn't make it obvious how to solve.
Just to be extra safe, I put it in a spoiler box, so that if you want, you'll only know it has relevance to one of the problems there, but not which one.
My proof looks like this :
I know this needs editing but I'm pretty sure it is a valid proof, probably formulated in a twisted way though. What do you think ?Originally Posted by Proof draft
Thanks for the link undefined, I'm bookmarking it.