Let $a_n$ denote the Fibonacci numbers, for a prime $p$ let $\alpha(p)$ denote the first index $n$ such that $p|a_n$ and let $r$ denote the golden ratio.

Q: Is there a proof of $\lim_{x\rightarrow \infty}\frac{1}{x^2}\sum_{\alpha(p)\leq x} \log p = \frac{3 \log r}{\pi^2}$ ?

The sum is taken over all primes $p$ that divide at least one Fibonacci number with index less or equal to $x$.

It is related to the number of Wall-Sun-Sun primes (cf. e.g. Is the Crandall, Dilcher and Pomerance heuristic concerning Wall-Sun-Sun primes still state of the art?).

Added later: It is known (P. Kiss, Primitive Divisors of Lucas Numbers, "Application of Fibonacci Numbers" (A. N. Phillipou et al.,Ed.), pp. 29 -- 38, Kluwer Acad. Publ. 1988) that $\limsup_{x\rightarrow \infty}\frac{1}{x^2}\sum_{\alpha(p)\leq x} \log p \leq \frac{3 \log r}{\pi^2}$.

  • $\begingroup$ You could start by looking up what is known about the "rank of apparition" of a prime $p$ in the Fibonacci numbers, if you haven't already. $\endgroup$ Jul 24, 2013 at 17:22
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    $\begingroup$ Possibly related is Matiyasevich and Guy, A new formula for $\pi$, Amer Math Monthly 93 (1986) 631-635, MR1712797 (2000i:11199), where it is proved that $$\pi=\lim_{n\to\infty}\sqrt{6\log a_1a_2\cdots a_n\over\log{\rm lcm}(a_1,a_2,\dots,a_n)}$$ $\endgroup$ Jul 25, 2013 at 3:36
  • $\begingroup$ @Greg: The $\alpha$ is indeed what some call the "rank of apparition". Since $d|a_n$ iff $\alpha(d)|n$ I prefer to call $\alpha$ "the dual sequence". This is no official term just less of a jawbreaker. $\endgroup$ Jul 25, 2013 at 12:56
  • $\begingroup$ @Gerry: Thanks a lot. Using the above as a representation of $\pi$ is somewhat obvious, but never came to my mind. The paper of Matiyasevich and Guy unfortunately is behind a pay wall. But now that I know what to search for I found Peter Kiss and Ferenc Matyas, An Asymptotic Formula for $\pi$, Journ. Numb. Th. 31, 255-259 (1989). They quote the paper of Matiyasevich and Guy. Their Lemma 2 (attributed to Kiss) is exactly why I ask this question. The most recent paper I found so far is Shigeki Akiyama, Lehmer Numbers and an Asymptotic Formula for $\pi$, Journ. Numb. Th. 36, 328-331 (1990). $\endgroup$ Jul 25, 2013 at 13:41
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    $\begingroup$ Akiyama had another paper, A new type of inclusion exclusion principle for sequences and asymptotic formulas for $\zeta(k)$, J. Number Theory 45 (1993), no. 2, 200–214, MR1242715 (94k:11027). I don't know whether you have come across Bogdan Tropak, Some asymptotic properties of Lucas numbers, in Proceedings of the Regional Mathematical Conference (Kalsk, 1988), 49–55, Pedagog. Univ. Zielona Góra, Zielona Góra, 1990, MR1114366 (92e:11013), or Jean-Paul Bézivin, Plus petit commun multiple des termes consécutifs d'une suite récurrente linéaire, Collect. Math. 40 (1989), no. 1, 1–11 (1990). $\endgroup$ Jul 25, 2013 at 23:36

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As far as I understand the answer to my question is no. That seems plausible since, differently from what I thought first, the truth of the conjecture apparently does not have serious consequences.


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