# Prime number and Relationship of Sequences of period 4,5,and 6

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

Let $$t^+,t^-,a$$ be sequences.
$$t^+_{k+5}=t^+_k,t^+_1=0,t^+_2=-1,t^+_3=-1,t^+_4=0,t^+_5=2$$
$$t^-_{k+5}=t^-_k,t^-_1=-1,t^-_2=0,t^-_3=0,t^-_4=-1,t^-_5=2$$
$$a_k=2(\cos\frac{k}{3}\pi-\cos\frac{k}{2}\pi)$$

Then,
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?

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

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