Let $p$ be a prime number.($p \neq 2,3,5$)

Let $t^+,t^-,a$ be sequences.

case $p\equiv1,4\pmod5$
$\underset{1\leq k\leq{p-1}}{\sum}\frac{a_k t^+_{p-k}}{k}\equiv0\pmod p$

case $p\equiv2,3\pmod5$
$\underset{1\leq k\leq{p-1}}{\sum}\frac{a_k t^-_{p-k}}{k}\equiv0\pmod p$

I have checked this for $p<10000$.
Can anyone prove this?

$t^+_k+t^-_k=e^{\frac{2}{5}k\pi i}+e^{\frac{4}{5}k\pi i}+e^{\frac{6}{5}k\pi i}+e^{\frac{8}{5}k\pi i}$

  • $\begingroup$ why do you use this notation ($t^+$, $t^-$) for your sequences? Why do you not use to different letters? $\endgroup$ – miracle173 Aug 19 at 7:55
  • $\begingroup$ Sorry. There is no particular meaning. $\endgroup$ – Takafumi Aug 19 at 9:04
  • $\begingroup$ from where do you have this sequence? $\endgroup$ – miracle173 Aug 19 at 12:35
  • $\begingroup$ In research on Fibonacci sequence. $\endgroup$ – Takafumi Aug 20 at 3:41
  • $\begingroup$ does k index only primes or index all positive integers $\endgroup$ – phdmba7of12 Aug 22 at 19:16

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