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It is evident from Fibonacci sequence mod(3) $$ 1,1,2,0,2,2,1,0,1,1,2,0,...$$ that every fourth term of the Fibonacci sequence is a multiple of $3$.

Similarly for mod(5) $$1,1,2,3,0,3,3,1,4,0,4,4,3,2,0,2,2,4,1,0,1,1,2,....$$ suggests that every fifth term in the Fibonacci sequence is a multiple of $5$.

It has been shown that every positive integer n divides the terms of the Fibonacci sequence periodically. The period depends on n .

Question: Is it possible to express the period p as a function of n? For example we have to get p(3)=4 and p(5)=5.

Mohammad Riazi-Kermani
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1 Answers1

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It is possible to show (with conventional numbering $F_1=1, F_2=1, F_3=2, F_4=3, F_5=5, F_6=8 \dots$) that $F_d|F_{nd}$ for any integer $n$.

The fact that the series recurs modulo any positive integer $r$ is inevitable because there are only finitely many pairs of integers mod $r$, and any pair determines the whole sequence.

I suggest you try to prove this divisibility rule - you might want to try showing $F_d|F_{2d}$ first, as this has some neat features which you might spot if you test the first few examples.

Mark Bennet
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  • Thanks for your constructive comment – Mohammad Riazi-Kermani Jan 06 '18 at 17:44
  • My question remains unanswered. Can we express p(n) as a function of n? – Mohammad Riazi-Kermani Jan 07 '18 at 01:48
  • @MohammadRiazi-Kermani I don't think it is easy to establish which $F_r$ is the first divisible by $n$ or even by a given prime $p$. The period for given $p$ is $r_p$ where $F_{r_p}$ is the first F_r divisible by $p$. We have $(p, r_p)= (2,3), (3,4), (5,5), (7,8), (11,10), (13,7), (17,9), (19,18),(23,24), (29,14) \dots (89,11) \dots$. Now there are some patterns to investigate. Aside from $5$ the period is a factor of $p\pm 1$, and the prime $5$ plays a special part here. – Mark Bennet Jan 07 '18 at 07:18
  • I am still working on it. Is 5 the only integer for which p(n)=n? For which integers p(n)<n, and many other questions. – Mohammad Riazi-Kermani Jan 07 '18 at 09:42
  • Is $F_d|F_{2d}$ true if d is not prime? – Mohammad Riazi-Kermani Jan 07 '18 at 09:55
  • @MohammadRiazi-Kermani Yes: F_{2d}=F_dL_d$ where $L_d$ is the related Lucas number. I think you would do well to do some reading about Fibonacci Numbers, and also some testing for yourself as to what patterns you can see. (I sometimes use a spreadsheet for such purposes, as it will tabulate values quite efficiently). I found https://en.wikipedia.org/wiki/Fibonacci_number, https://math.stackexchange.com/questions/511379/divisibility-of-fibonacci-numbers, https://en.wikipedia.org/wiki/Fibonacci_prime all of which have comments relevant to your question. – Mark Bennet Jan 07 '18 at 12:16
  • Thanks again. I will read more about Fibonacci Numbers and see if can figure out a formula for p(n). – Mohammad Riazi-Kermani Jan 07 '18 at 14:17