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Hi I need help on this question:

Consider the sequence of positive integers an, for $n ≥ 1$, defined by $a_n = 6^{2^n} + 1$.

(a) Prove that the elements of this sequence are pairwise co-prime, i.e. prove that if $m$ is not $n$ then $gcd(a_m, a_n) = 1$.

(b) Show how this result, combined with the Fundamental Theorem of Arithmetic, provides another proof that there are an infinite number of primes.

Hint: Begin the first part by proving that $a_n$ $|$ $(a_{n+1} − 2)$

I've tried proving that $a_n$ is a factor of $(a_{n+1} − 2)$ but now im not sure how to carry on to prove the $gcd$ is $1$.

anna
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iffy
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    $(a)$ is a special case of the first linked dupe, and $(b)$ follows by Euclid's method, as explained in my answer in the second linked dupe. – Bill Dubuque Feb 11 '21 at 19:49
  • See also the Remark in [this answer](https://math.stackexchange.com/a/68666/242) for the Hint. – Bill Dubuque Feb 11 '21 at 19:54

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