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Let $k$ be a fixed integer and for any prime number $p$ larger than $k$, let $\text{Order}(k,p)$ be the order of $k$ in $\mathbb{F}_p$ (i.e., $\text{Order}(k,p)$ is the least integer $n$ such that $k^n \equiv 1 \bmod{p}$).

Question. Give an upper bound to the sequence $$v_n=\inf_{t>n}\left(\text{Order}(k,p_t)\right),$$ where the $p_t$'s are the enumeration of all prime numbers in increasing order.

Fermat's little theorem provides the bound $v_n\leq p_n - 1$. Can a bound of $p_n^\epsilon$ be shown, for any $\epsilon>0$? Experiments suggest a bound of $\log(p_n)$.

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This is not an answer, but maybe it's useful.

For example, take $k=2$. If $2^q-1$ is a Mersenne prime, it is $p_m$ where $q \sim \log_2 m$. Of course $\text{Order}(2, 2^q-1) = q$, so $v_n \le q$ if $n \le m$. If there are infinitely many Mersenne primes we will have $$\liminf_{n \to \infty} \frac{v_n}{\log_2(n)} \le 1$$

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    $\begingroup$ Yes, actually up to the generalized Mersenne conjectures you could have iterated log. Although I believed it not equivalent to such a conjecture, but I don't have any insight. It doesn't help for $k\neq 2$ though. $\endgroup$
    – C.P.
    Apr 5, 2019 at 19:15
  • $\begingroup$ For $k \ne 2$ you can consider generalized repunit primes: primes of the form $(k^n-1)/(k-1)$ $\endgroup$ Apr 5, 2019 at 22:51
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Rephrasing, you are calling $ v_n $ the smallest integer $m$ such that $ k^m - 1$ has a prime factor not among the first $n$ enumerated primes.

If you have the standard numbering (primes in increasing order), then $v_n$ is going to look like $m$ or less right up until you hit the largest prime factor of $k^m - 1$, which presumably is larger than any prime factor of the same form for smaller values of $m$. Since these primes are often large, this accords with your observed values. This is because prime factors of $k^m - 1$ as $m$ varies are often larger than a fixed power (expect at least one of them to be larger than $k^{m/2} -1$).

However, let's pretend you pick the unluckiest enumeration, so that the first n primes are the factors of the first m numbers of the form $k^m - 1$. Zsigmondy says that for each m, a new prime appears as a factor of $k^m-1$ (unless m=6 and k=2). So $v_n$ will eventually be less than $n$, and conjecturally less than $n/log n$, even if you are unlucky.

Gerhard "The Fortunes Of Number Theory" Paseman, 2019.04.05.

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  • $\begingroup$ Thank you for your comment, I have edited my question since I only care about standard numbering. I don't really understand your rephrasing. What is the link between the fact that $k^m-1$ has a prime factor which not among the first $n$ enumerated primes and the evolution of the $\text{Order}(k,p_i)$ sequence? $\endgroup$
    – C.P.
    Apr 6, 2019 at 15:09
  • $\begingroup$ If k has order m in a field of characteristic p (so k is not zero mod p), then p divides k^m - 1 and for n smaller than m p does not divide k^n - 1. Since you are doing inf over large primes, for primes q smaller than the largest prime p_j dividing any k^n - 1 up to k^m - 1, all v_i for i up to j will be m or less. j is usually like exp(m), as you have observed. Gerhard "Will Do The Slow Climb" Paseman, 2019.04.06. $\endgroup$ Apr 6, 2019 at 18:45
  • $\begingroup$ I do believe it is true, but how can we prove that $j$ is usually $e^m$ ? $\endgroup$
    – C.P.
    Apr 7, 2019 at 12:09
  • $\begingroup$ I would look at prime values of m and try to get a result that says that a large prime divisor exists among the set of k^m - 1 where prime m comes from a small interval. Right now, the best results talk about factors of cyclotomic polynomials for a single exponent, but you don't need such a strong requirement: you just need a nice sprinkling of large factors. Gerhard "To Go With The Frosting" Paseman, 2010.04.07. $\endgroup$ Apr 7, 2019 at 15:50

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