How to prove $641|2^{32}+1$

Question

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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. :-)

This has already been asked in this math.SE question (and therefore should be closed as a duplicate). — Zev Chonoles, Jun 08 '12 at 18:14
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, Jun 08 '12 at 18:26

score 7 · Accepted Answer · edited Jul 05 '13 at 06:54

7

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.

edited Jul 05 '13 at 06:54

Martin Sleziak

answered Jun 08 '12 at 18:33

Mikasa

1 Answers1