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LDP

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Let F

We know that F

Let

Let

If

Then:

and

Now let

If

Note that

If

Let

and

If

_{n}be the n^{th}number of a Fibonacci sequence.We know that F

_{n}mod(*p*) forms a periodic sequence (http://en.wikipedia.org/wiki/Pisano_period) called the Pisano Period.Let

*p*= a prime such that*p*[itex]\equiv[/itex]{2,3}mod 5 so that*h*(*p*)[itex]\mid[/itex] 2*p*+ 2.Let

*h*(*p*) denote of the length of the Pisano period.If

*D*= {*d*,_{1}*d*,_{2}*d*[itex]\cdots[/itex]_{3}*d*} is the non-empty set of_{k}*k*divisors of 2*p*+ 2Then:

*h*(*p*) = min[*d*] such that F_{i}_{d(i + 1)}[itex]\equiv[/itex] 1 mod*p*and

*d*~[itex]\mid[/itex][itex]\frac{1}{2}[/itex]_{i}*p*(*p*+ 1)*d*~[itex]\mid[/itex]_{i}*p*+ 1*d*~[itex]\mid[/itex] 3 (_{i}*p*- 1)

Now let

*p*= a prime such that*p*[itex]\equiv[/itex]{1,4}mod 5 so that*h*(*p*)[itex]\mid[/itex]*p*- 1.If

*p*has a primitive root such that*g*[itex]\equiv[/itex]^{2}*g*+ 1 mod(*p*) then*h*(*p*) =*p*- 1.Note that

*g*[itex]\equiv[/itex]^{2}*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*= {*d*,_{1}*d*,_{2}*d*[itex]\cdots[/itex]_{3}*d*} is the non-empty set of_{k}*k*divisors of*p*- 1.Let

*h*(*p*) = min[*d*] such that F_{i}_{d(i + 1)}[itex]\equiv[/itex] 1 mod*p*and

*d*~[itex]\mid[/itex]_{i}*p*+ 1 and*d*~[itex]\mid[/itex]_{i}**floor**[*p*/2]].If

*m*is any positive integer > 3 we can write F*mod F*_{n}*where*_{m}*h*(F*) is given by*_{m}-
*h*(F) = 2_{m}*m*↔*m*is even *h*(F) = 4_{m}*m*↔*m*is odd

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