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 2007-06-28, 00:20 #1 vtai   Jun 2007 310 Posts help with a proof let p be a prime. show that 2^Mp = 2(mod Mp). (where Mp is the Mersenne number for p). i was able to show that the congruence is true for fermat numbers (ie. 2^Fn = 2(mod Fn)). but i just can't seem to figure it out for this question. any help with this proof would be appreciated.
 2007-06-28, 11:11 #2 paulunderwood     Sep 2002 Database er0rr 2×72×37 Posts
2007-06-28, 11:19   #3
R.D. Silverman

Nov 2003

22·5·373 Posts

Quote:
 Originally Posted by paulunderwood http://primes.utm.edu/glossary/page....sLittleTheorem
And? Note that while p is prime, M_p usually isn't. We are working mod
M_p, not mod p.

 2007-06-28, 12:22 #4 alpertron     Aug 2002 Buenos Aires, Argentina 25048 Posts Let p be a prime number. $2^p \equiv 1 \,\pmod {2^p-1}$ Using Fermat's Little Theorem we get: $2^p \equiv 2 \,\pmod p$ so $2^p-1 \equiv 1 \,\pmod p$ This means that $2^p-1 = kp+1$ so we finally get: $2^{2^p-1} \equiv 2^{kp+1} \equiv 2^{kp}*2^1 \equiv (2^p)^k*2 \equiv 1^k*2 \equiv 2\,\pmod {2^p-1}$ Last fiddled with by alpertron on 2007-06-28 at 12:28
 2007-06-28, 12:23 #5 R. Gerbicz     "Robert Gerbicz" Oct 2005 Hungary 1,459 Posts Mp=2^p-1, where p is prime. By Fermat's Little Theorem 2^p==2 mod p, so Mp==1 mod p, so Mp=k*p+1 for some positive integer k, using this: 2^Mp=2^(k*p+1)=2*2^(k*p)=2*(2^p)^k=2*(Mp+1)^k==2*1^k==2 mod Mp, what is needed.
2007-06-28, 12:37   #6
paulunderwood

Sep 2002
Database er0rr

2×72×37 Posts

Quote:
 And?
As this is not the homework section...

Quote:
 Note that while p is prime, M_p usually isn't. We are working mod M_p, not mod p.
Thanks for pointing this out, Bob.

2^p==2 (mod p) by Fermat's Little Theorem since gcd(2,2^p-1)==1
2^p-2==0 (mod p) by subtracting "2" from each side of the congruence equation

By the definition, p divides 2^p-2. That is 2^p-2 is a multiple of p. So 2^p-2 =N*p for some "N".
2^p-1=N*p+1 by adding "1" to each side of the equation.

2^p-1==0 (mod Mp) by definition
2^p==1 (mod Mp) by adding "1" to each side of the congruence equation[*]

Therefore, 2^Mp==2^(N*p+1) (mod Mp)
2^Mp==(2^(N*p))*2 (mod Mp)
2^Mp==((2^p)^N)*2 (mod Mp)
2^Mp==(1^N)*2 (mod Mp) by[*]
2^Mp==2 (mod Mp)

Q.E.D.

This demonstrates why a base 2 Fermat test is useless for Mersenne primes.

(I see that by the time I have written this out two other people have given the proof )

Last fiddled with by paulunderwood on 2007-06-28 at 13:12

2007-06-28, 13:35   #7
alpertron

Aug 2002
Buenos Aires, Argentina

54416 Posts

Quote:
 Originally Posted by paulunderwood 2^p==2 (mod p) by Fermat's Little Theorem since gcd(2,2^p-1)==1
It should be ...since gcd(2,p)==1

 2007-06-28, 13:50 #8 paulunderwood     Sep 2002 Database er0rr 2·72·37 Posts
 2007-06-28, 14:04 #9 vtai   Jun 2007 38 Posts Thank your for your replies...
 2007-06-28, 14:36 #10 vtai   Jun 2007 3 Posts i have another question that is related...what if p was not prime, how would we prove the congruence then??
2007-06-28, 14:50   #11
alpertron

Aug 2002
Buenos Aires, Argentina

22×337 Posts

Quote:
 Originally Posted by vtai i have another question that is related...what if p was not prime, how would we prove the congruence then??
In that case in general $2^p\not\equiv 2\,\pmod p$, so you will not get the number 2.

But since $2^p \eq 1\,\pmod{2^p-1}$ at least you always get a power of 2.

For example if p=6 you get:
$2^{2^p-1} \eq 2^{2^p-1 \bmod p} \eq 2^{63 \bmod 6} \eq 2^3 \eq 8\,\pmod{2^p-1}$

Last fiddled with by alpertron on 2007-06-28 at 15:02

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