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What is the units digit of $2^{2^{1000}}+1$(The $1000^{th}$ Fermat prime)?

 Apr 5, 2021
 #1
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Note that $2^{13}\equiv 2^1\pmod{10}$ so $2^{n}=2^{n+12}\pmod{10}$. Then $2^{1000}\equiv 4\pmod{12}$ so $2^{2^{1000}}\equiv 2^4\equiv 6\pmod{10}$. The answer is $6+1=7$.

 Apr 5, 2021
 #2
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Thank you, it was correct!

RiemannIntegralzzz  Apr 5, 2021

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