### fermat's number

Posted:

**Tue Dec 14, 2010 2:16 pm**Prove that 5th Fermat's number $2^{2^5}+1$ is a multiple of 641.

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Posted: **Tue Dec 14, 2010 2:16 pm**

Prove that 5th Fermat's number $2^{2^5}+1$ is a multiple of 641.

Posted: **Tue Dec 14, 2010 2:44 pm**

i tried to write 2^2^5 as dollar($){2^(2^5)+1}dollar($) but didn't worked!!!!!!!!!!someone help me to write like that......

about problem:i used modular and after about restless 8 days i made the solution of it.........now after yours solution I'll post my one............it was the hardest problem I'd ever solved..........

about problem:i used modular and after about restless 8 days i made the solution of it.........now after yours solution I'll post my one............it was the hardest problem I'd ever solved..........

Posted: **Tue Dec 14, 2010 3:13 pm**

You have towrite 2^{2^5}+1 (I think you made mistake last time)

I mean when you want to write something bigger than 1 letter you have to use second bracket.

So, 2^{2^5}+1=$2^{2^5}+1$

I mean when you want to write something bigger than 1 letter you have to use second bracket.

So, 2^{2^5}+1=$2^{2^5}+1$

Posted: **Wed Dec 15, 2010 11:56 am**

This is due to Euler.

Note that $641=5^4+2^4|2^{32}+5^42^{28}$ and $641=5.2^7+1|5^42^{28}-1$

So $641$ divides their difference $2^{32}+1$

Note that $641=5^4+2^4|2^{32}+5^42^{28}$ and $641=5.2^7+1|5^42^{28}-1$

So $641$ divides their difference $2^{32}+1$

Posted: **Wed Dec 29, 2010 12:19 pm**

Ok, this is an easy problem..if you read in wikipedia about FERMAT NUMBER you can get the answer........2^2^5=2^32=4294967297

so ,you can see that 4294967297/641...........and that is your ans.

so ,you can see that 4294967297/641...........and that is your ans.

Posted: **Wed Dec 29, 2010 1:34 pm**

i dont think you will have access to wikipedia or calculator in any MATH contest .Dipan wrote:Ok, this is an easy problem..if you read in wikipedia about FERMAT NUMBER you can get the answer........2^2^5=2^32=4294967297

so ,you can see that 4294967297/641...........and that is your ans.

Posted: **Wed Dec 29, 2010 1:47 pm**

But this is not allowed in the olympiad and you need to prove this rigorouslyDipan wrote:Ok, this is an easy problem..if you read in wikipedia about FERMAT NUMBER you can get the answer........2^2^5=2^32=4294967297

so ,you can see that 4294967297/641...........and that is your ans.

And we need the proof,you just checked this.

Posted: **Wed Dec 29, 2010 10:31 pm**

Sorry..