2
votes
1answer
32 views

Is a Lucas Number with either a power of 2 or a prime index always coprime with all previous Lucas Numbers?

I was looking at this webpage which lists the first 200 Lucas Numbers color-coded with their prime factors and I noticed that all the Lucas numbers with power of two or prime indexes were relatively ...
1
vote
1answer
41 views

Does this mean some Wall-Sun-Sun primes have already been found?

In the PrimeGrid project statistics page for Wall-Sun-Sun Prime Search, it says, Wall-Sun-Suns ... 2 Near Wall-Sun-Suns ... 208 However, all the internet search ...
40
votes
5answers
4k views

Does every prime divide some Fibonacci number?

I am tring to show that $\forall a \in \Bbb P\; \exists n\in\Bbb N : a|F_n$, where $F$ is the fibonacci sequence defined as $\{F_n\}:F_0 = 0, F_1 = 1, F_n = F_{n-1} + F_{n-2}$ $(n=2,3,...)$. How can ...
-2
votes
1answer
67 views

Application of convergence of Fibonacci series

'There are infinite prime numbers' is a fact that can be deduced by 'reciprocal of primes diverges' statement, so from this can we deduce the fact that --> 'there are finite Fibonacci numbers in ...
1
vote
1answer
232 views

Pisano periods of fibonacci mod

The wikipedia article on Pisano periods utilises the Binet's formula and quadratic residues to find $f(n)$ such that $F_n=f(n) \pmod{p}$ where $p$ is a prime number and $F_n$ is a Fibonacci number. ...
0
votes
2answers
211 views

find a general expression for the remainder when a prime divides a fibonacci.

I have primes of form $5k\pm1$. Consider the equation: $F_n=f(n)\pmod p$ where $F_n$ is the nth fibonacci number. Now given a c, how can i check whether or not there exists a solution for $f(n)=c ...
0
votes
0answers
227 views

fibonacci numbers mod some prime number

Moderator Note: This is a current Code Chef challenge question. When the current challenge ends on 15 October 2013 this question will be unlocked. I have prime numbers ($\geq11$) and of the form ...
0
votes
0answers
82 views

Fibonacci sequense, problem od division

How to show that $7\mid F_m\Longrightarrow 8\mid m$ and $4\mid F_m\Longrightarrow 6\mid m$, knowing that (I) Two consecutive terms in the Fibonacci sequence are relatively prime. (II) In ...
6
votes
1answer
228 views

Let $F_n$ be a Fibonacci number and $p$ a prime. Verify that for $p \le 61$, if $p\equiv\pm1 \pmod{5}$ then $p\mid F_{p-1}$

Define the Fibonacci entry point of $p$ to be the least integer $n$ such that $p\mid F_n$ So for example, for $p = 3$ - the Fibonacci entry point is $n = 4$ since $F_4 = 3$ and obviously $3\mid 3$. ...
8
votes
1answer
91 views

Prove that $\forall p \in \Bbb P;p \ne 5,$ $F_{p^n - \left(\frac{5}{p}\right)p^{n-1}} \equiv 0 \mod p^n$

Prove that $\forall p \in \Bbb P,n \in \Bbb Z^+;p \ne 5,$ $F_{p^n - \left(\frac{5}{p}\right)p^{n-1}} \equiv 0 \mod p^n$ and $F_{5^n} \equiv 0 \mod 5^n$, where $\left(\dfrac{5}p\right)$ is the Legendre ...
3
votes
0answers
130 views

Which starting conditions for the Fibonacci sequence, gives most primes

I found the following question (at http://aperiodical.com/2012/05/matt-parkers-twitter-puzzle-25-may/): If you start the Fibonacci sequence 2,1 instead of 1,1 do you get more or fewer primes? ...
4
votes
1answer
296 views

What is the next “Tribonacci-like” pseudoprime?

Given the three roots of $x^3=x^2+x+1$. Then we get the tribonacci-like sequence, $B_n = x_1^n+x_2^n+x_3^n = 3, 1, 3, 7, 11, 21, 39, 71, 131,\dots$ where $B_n = B_{n-1}+B_{n-2}+B_{n-3}$, and the ...
2
votes
1answer
376 views

Prime power divisors of the fibonacci numbers

I came across a result that if $p^n \mid f_m$ for some $n\geq1$ then $p^{n+1} \mid f_{pm}$. I was wondering if this is true.
6
votes
4answers
328 views

Prime Appearances in Fibonacci Number Factorizations

Okay, THIS one is considerably more analytical... :P (Used my post here as a basis.) When successive Fibonacci numbers are factored, the primes appear in a specific order, which goes $2, 3, 5, 13, 7, ...
4
votes
1answer
124 views

Prove that If $f_n$ where $n>3$ is prime, then $n$ is prime for a Fibonacci series where $f_1$=$f_2$=1

This problem came up in my conversation with a friend—not sure how basic it is, but it seems quite interesting: Prove that if $f_n$ where $n>3$ is prime, then $n$ is prime for a Fibonacci sequence ...