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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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marked as duplicate by Asaf Karagila, Arturo Magidin, Ross Millikan, Byron Schmuland, Zev Chonoles Jun 8 '12 at 19:27

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

    
This has already been asked in this math.SE question (and therefore should be closed as a duplicate). –  Zev Chonoles Jun 8 '12 at 18:14
    
@ZevChonoles: I didn't know that. Thank you. –  Basil R Jun 8 '12 at 18:17
4  
In a nutshell. Look at the congruences: $$\eqalign{ & {2^{16}} \equiv 65536 \equiv 154\bmod 641 \cr & {2^{32}} \equiv {154^2} \equiv - 1\bmod 641 \cr & {2^{32}} + 1 \equiv 0\bmod 641 \cr} $$ –  Pedro Tamaroff Jun 8 '12 at 18:26

1 Answer 1

up vote 4 down vote accepted

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.

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Nicely done $+1\quad \ddot\smile\quad$ –  amWhy Mar 6 '13 at 0:50

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