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Question Number 180211 by Shrinava last updated on 09/Nov/22

Find:  Ω = gcd (4^(2020) −1 , 8^(2021) −1). Generalization.

$$\mathrm{Find}: \\ $$$$\Omega\:=\:\mathrm{gcd}\:\left(\mathrm{4}^{\mathrm{2020}} −\mathrm{1}\:,\:\mathrm{8}^{\mathrm{2021}} −\mathrm{1}\right).\:\mathrm{Generalization}. \\ $$

Answered by puissant last updated on 09/Nov/22

Ω = gcd(2^(4040) −1 ; 2^(6063) −1)       = 2^(gcd(4040 ; 6063)) −1       = 2^1 −1       = 1.

$$\Omega\:=\:{gcd}\left(\mathrm{2}^{\mathrm{4040}} −\mathrm{1}\:;\:\mathrm{2}^{\mathrm{6063}} −\mathrm{1}\right) \\ $$$$\:\:\:\:\:=\:\mathrm{2}^{{gcd}\left(\mathrm{4040}\:;\:\mathrm{6063}\right)} −\mathrm{1} \\ $$$$\:\:\:\:\:=\:\mathrm{2}^{\mathrm{1}} −\mathrm{1} \\ $$$$\:\:\:\:\:=\:\mathrm{1}. \\ $$

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