13

I am looking for references to the following problem, which I saw a long time ago and I think is a well-known problem (maybe from IMO or American Mathematical Monthly), I hope to remember it correctly.

Problem. Let $a$ and $b$ be two positive integers. If $a^n - 1$ divides $b^n - 1$ for all the positive integers $n$, then $b = a^k$ for some positive integer $k$.

Thank you in advance for any answer.

EDIT: As noted by cr001, a solution to the problem was given on Math.StackExchange (see question 417340). However, what I am asking for is a reference to a journal or book.

Community
  • 1
  • Did you make sure that this is not a duplicate ? – Elaqqad Oct 04 '15 at 10:38
  • @Elaqqad duplicate of what? I didn't find the same question on math.stackexchange, pearphs there is, but I don't know. –  Oct 04 '15 at 13:56
  • 1
    I found a reference on this forum actually. The question is different but the answer solves the exact problem. (And I removed my incomplete solution as well, if anyone appears to work on my solution and managed to prove it please post a new answer.)

    http://math.stackexchange.com/questions/417340/do-there-exist-two-primes-pq-such-that-pn-1-mid-qn-1-for-infinitely-many?rq=1

     – cr001 Oct 05 '15 at 09:17
  • @cr001 Thank you, I'm gonna add the link you pointed out in my question. –  Oct 05 '15 at 16:28
  • It's known that if $b\neq a^k$ for every $k$ then $a^n-1$ can not divide $b^n-1$ infinitely often , as a reference you may look for "Hadamard Quotient Theorem" or some of it's generalizations which are transferred to the Diophantine equations – Elaqqad Oct 05 '15 at 19:58
  • @Elaqqad You are right about Hadamard Quotient Theorem, but I am asking for a reference to the original problem with $a^n-1$ and $b^n-1$. –  Oct 05 '15 at 20:10
  • 1
    Here is a reference for my claim "but not the original one" (I did not find the original) see page $427-\cdots$, The proof is not quite elementary it uses some analysis and a well known theorem due to Wolfgang M. Schmidt called Subspace theorem – Elaqqad Oct 06 '15 at 07:40
  • Your title seems subjective. What is your definition of "too often?" – Obinna Nwakwue Jun 04 '17 at 21:08

0 Answers0