TITLE: Prove that Fibonacci sequence contains An infinite Sub-sequence That all its element are co-prime.
QUESTION [6 upvotes]: Prove that Fibonacci sequence contains An infinite Sub-sequence That all its element are co-prime.
Prove Two Sub-sequence different

Instruction: Fermat numbers.

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attempt 1:
$F_n$ is strong divisibility sequence , $\gcd(F_m,F_n)$=$F_{\gcd(m,n)}$
we need to prove there an infinite sequence of coprime integers.

REPLY [3 votes]: Hint: by the simple proof in this answer, the Fibonacci sequence $\,F_n\,$ is strong divisibility sequence, i.e. $\,\gcd(F_m,F_n) = F_{\gcd(m,n)},\,$ so the problem reduces to proving that there is an infinite sequence of coprime integers (indices), for which there are many obvious methods (some that don't require primes, e.g. Euclid's famous constructive algorithm that recursively generates an infinite sequence of (co)prime integers).