Let $p$ be an odd prime and $L_n$ be the $n$th Lucas number. Can anyone prove this? $$\frac{L_1}{1}+\frac{L_3}{3}+\frac{L_5}{5}+\cdots+\frac{L_{p-2}}{p-2}\neq0\pmod{p}$$ Please help me!

I am thinking about the period of Fibonacci sequence. The purpose is to prove that (the period mod $p$) $\neq$ (the period mod $p^2$). It is known that the period mod $p$ divides $p-1$ or $2p+2\ (p\neq5)$. So, I tried to prove $F^{p^2-1}\neq I\ \pmod{p^2}$, where $F$ is the Fibonacci matrix, then I got this inequality.

If $p\neq5$, this question is equivalent to prove that $$\frac{\phi}{1}+\frac{\phi^2}{2}+\frac{\phi^3}{3}+\cdots+\frac{\phi^{p-1}}{p-1}\neq0\pmod{p}.$$ Here, $\phi$ is a root of $x^2-x-1$. See this page. I have showed that $$\frac{\phi}{1}+\frac{\phi^2}{2}+\frac{\phi^3}{3}+\cdots+\frac{\phi^{p-1}}{p-1}\equiv\sum_{k=1}^{\frac{p-1}{2}} \frac{{(-1)}^k}{2k} {{2k}\choose{k}}\pmod{p}.$$

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  • Welcome to MSE! I would like to mention that providing context or your own thoughts on the question is highly recommended. You may give a brief look on [*this posting*](https://math.meta.stackexchange.com/questions/9959/how-to-ask-a-good-question#9960) to see what may improve your question. – Sangchul Lee Jan 06 '18 at 04:49
  • Thank you for your advice. – Takafumi Jan 06 '18 at 05:13
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    Watch out: the order of $10 \bmod p$ and $10 \bmod p^2$ are equal when $p = 3$, $487$, and $56598313$. Look up base-$a$ Wieferich primes. – KCd Jan 06 '18 at 06:56
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    My previous comment was just pointing out that some analogous problems mod $p$ and $p^2$ can have sporadic counterexamples. I see from https://arxiv.org/pdf/1511.01210.pdf that what you are trying to show is called Wall's conjecture, but given the similarity to Wieferich primes I am dubious about the lack of counterexamples up to big bounds really being suggestive of there not being any counterexample at all. – KCd Jan 06 '18 at 07:04
  • Using some identities from an answer to [this question](https://math.stackexchange.com/questions/2613507), for $p\equiv\pm1\pmod5$ the congruence is _equivalent_ to $p$ not being a counterexample to Wall's conjecture. Can't tell for the other $p$'s. – Bart Michels Jan 25 '18 at 22:28

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