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Dear all,

I have a problem relating to primitive prime divisors. (For a sequence $(A_n)_{n\ge1}$, a primitive prime divisor of a term $A_n$ is a prime $\ell$ that divides $A_n$, but does not divide $A_i$ for all $i<n$.)

Assume that $p$ and $q$ are distinct odd primes and $n\geq 3$ a fixed positive integer, so that $p^n-1$ and $q^n-1$ have at least one primitive prime divisor (by the Bang-Zsigmondy theorem).

  • Question 1: When do $p^n-1$ and $q^n-1$ have a common primitive prime divisor?

  • Question 2: When do $p^n-1$ and $p^n-q^n$ have a common primitive prime divisor (which will be a divisor, although not necessarily primitive, of $q^n-1$)?

If you have some source, please tell me.

Thanks a lot in advance.

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  • $\begingroup$ Your question is ambiguous because $p_n$ is not uniquely determined. $\endgroup$
    – Felipe Voloch
    Commented Mar 17, 2012 at 1:40
  • $\begingroup$ Thank Felipe. Yes, $p_n$ is not unique, however, I just call here $p_n$ for one of those. $\endgroup$
    – Ben
    Commented Mar 17, 2012 at 7:18
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    $\begingroup$ @Dung Duong: I reworded your question to try to make it clearer. I hope that I haven't changed what you want to know. $\endgroup$
    – Joe Silverman
    Commented Mar 17, 2012 at 12:20
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    $\begingroup$ Try checking Ribenboim's book "My Numbers, My Friends". It's very much a survey, but if there's something to be said (i.e. if these are not open questions), it has the source. Your sequences are $A_0 = 0$, $A_1 = p-1$, and $A_n = (p+1)A_{n-1} - pA_{n-2}$, so covered by general linear recurrences. $\endgroup$
    – Zack Wolske
    Commented Mar 17, 2012 at 22:25

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