Question

Find the greatest comman divisor of
$3 {}^{3 {}^{333} } + 1 \: and \: 3 {}^{3 {}^{334} } + 1$
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Answer 1

Since a³+1=(a+1)(a²−a+1) , gcd(a+1,a3+1)=a+1 .

Now with a=3³³³³ , a³=(33333)3=33333⋅3=3³³³⁴ . Hence the gcd equals 33333+1 . ■