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Possible Duplicate:
To show that Fermat number $F_{5}$ is divisible by $641$.

How to prove that $641$ divides $2^{32}+1$? What the technical way will be for this question? I want to teach it to my students. Any help. :-)

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    In a nutshell. Look at the congruences: \eqalign{ & {2^{16}} \equiv 65536 \equiv 154\bmod 641 \cr & {2^{32}} \equiv {154^2} \equiv - 1\bmod 641 \cr & {2^{32}} + 1 \equiv 0\bmod 641 \cr} 2012-06-08

1 Answers 1

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In light of Peter's comment:

we have:

$2^2=4$,

$2^4=16, 2^8=256,$

$2^{16}=256^2=65536=641k_1+154,$

$2^{32}=641k_2+154^2=641k_3+640$

the rest is very easy.