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I recently asked a question on the upper bound of the maximum length of a string of co-prime numbers in Fibonacci sequence. The answer appears to be 5. From the comments that lulu made on that question, it appears that a string of 5 occurs after a multiple of 6 where a multiple of 6 occurs at an interval of 24 terms starting from the first term. I calculated this for a tiny data set but this seems to be true. So, can some rigorously prove or disprove this claim? Proving this seems beyond me now (but perhaps we can make some clever claims?)

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    Are you aware that $\gcd(F_a, F_b ) = F_{\gcd(a, b) } $? This question essentially just uses that (and so it might be closed as an abstract duplicate, which is why I'm not posting this as an answer. If this counts as a hint and the comment should be deleted, let me know how best to proceed). $\quad$ Your claim is true. Using the fact stated above, you want terms whose GCD is either 1 or 2 only, and this is achieved IFF with the integers $ 6n - 2, 6n-1, 6n, 6n+1, 6n+2$. – Calvin Lin Mar 03 '25 at 18:39
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    Edit to my initial comment needed: My classification is wrong as I was misled by OP's claims. For the middle number $F_k$, we want $ k \equiv 0 \pmod 3$ and $ k \equiv 0, 1, 3 \pmod{4}$. This gives $ k \equiv 3, 9, 0 \pmod{12}$. EG With $F_9 = 34$, we have $13, 21, 34, 55, 89. $ – Calvin Lin Mar 03 '25 at 18:59
  • The Pisano period of the fibonacci numbers modulo 6 is not 12 but 24, see https://oeis.org/A001175 . – R. J. Mathar Mar 04 '25 at 16:38

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