Pisano Periods - Fibonacci Numbers mod p

1. Feb 3, 2013

LDP

Let Fn be the nth number of a Fibonacci sequence.
We know that Fnmod(p) forms a periodic sequence (http://en.wikipedia.org/wiki/Pisano_period) called the Pisano Period.
Let p = a prime such that p$\equiv${2,3}mod 5 so that h(p)$\mid$ 2 p + 2.
Let h(p) denote of the length of the Pisano period.

If D = {d1,d2,d3$\cdots$dk} is the non-empty set of k divisors of 2 p + 2
Then:
h(p) = min[di] such that Fd(i + 1)$\equiv$ 1 mod p
and
1. di ~$\mid$$\frac{1}{2}$ p (p + 1)
2. di ~$\mid$ p + 1
3. di ~$\mid$ 3 (p - 1)

Now let p = a prime such that p$\equiv${1,4}mod 5 so that h(p)$\mid$ p - 1.
If p has a primitive root such that g2$\equiv$ g + 1 mod(p) then h(p) = p - 1.
Note that g2$\equiv$ g + 1 mod(p) has two roots: 1.618033988 and -0.618033988 - variants of the Golden Ratio.
If p has no primitive root then D = {d1,d2,d3$\cdots$dk} is the non-empty set of k divisors of p - 1.
Let h(p) = min[di] such that Fd(i + 1)$\equiv$ 1 mod p
and
di ~$\mid$ p + 1 and di ~$\mid$ floor [ p/2]].

If m is any positive integer > 3 we can write Fn mod Fm where h(Fm) is given by
1. h(Fm) = 2mm is even
2. h(Fm) = 4mm is odd

Last edited: Feb 4, 2013
2. Feb 10, 2013

ramsey2879

Interesting to say the least. I am curious as to what is meant by
1. di ~$\mid$$\frac{1}{2}$ p (p + 1)
2. di ~$\mid$ p + 1
3. di ~$\mid$ 3 (p - 1)
I think you are saying di approximately divides the expressions on the right, but I dont know what that means. Can you give an example?

3. Feb 10, 2013

LDP

Thanks.

No, I was trying to find - does not divide - but couldn't so I sort of made that up.
But perhaps I should clarify it, because it is not the standard notation.

4. Feb 10, 2013

ramsey2879

Clarification noted. Only problem that I see is for p = 2 (for which you say h(p) = 2p + 2 = 6). Since h(2) = 3, I guess you meant odd primes that = 2,3, mod 5 (whose last digit is either a 3 or 7). PS, I noted that for the composites ending in 3 or 7 which I checked, that h(p) <> 2p + 2. Could this be a test for primes ending in 3 or 7?

5. Feb 11, 2013

LDP

Indeed, I should have said "only odd primes".
Good catch.
I will add some more observations on this topic as time permits.

6. Feb 12, 2013

ramsey2879

I checked and found that the following composites ending in 7 have Pisano periods that divide 2p + 2 and do not divide (1/2)*p(p+1) or p+1 or 3p-3. (All primes ending in 7 less than 100,000 meet this test also). The composites ending in 7 and less than 100,000 meeting the test are 377, 3827, 5777, 10877, 25877, 60377, 75077 and 90287.