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 2012-09-19, 16:06 #34 alpertron     Aug 2002 Buenos Aires, Argentina 5·269 Posts Based on the previous post: let n = (22p -2p+1)/3 where p is a prime number, is it true that 2n-1 = 1 (mod n) ?
2012-09-19, 16:55   #35
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

2×727 Posts

Quote:
 Originally Posted by alpertron Based on the previous post: let n = (22p -2p+1)/3 where p is a prime number, is it true that 2n-1 = 1 (mod n) ?
Nice problem. Proof: let p>2 prime number

Observe that: (2^(2*p)-2^p+1)/3 | 2^(6*p)-1
so it is enough to prove that for p>2: 2^(n-1)==1 mod 2^(6*p)-1
it is equivalent to n-1==0 mod (6*p)
so (2^(2*p)-2^p-2)/3==0 mod (6*p)
(18*p)|2^p*(2^p-1)-2

Let p>3 then it is true mod p (use Fermat's little theorem), 2 also divides, this is trivial since 2^p is even. For 9 use: p=6k+-1 then 2^p=={2,5} mod 9, for the two cases: 2^p*(2^p-1)-2=={0,18}==0 mod 9.
Since 2,9,p are relative prime numbers for p>3 it means that the divisibility is also true for 2*9*p=18*p.
For p=3: 2^(n-1)==1 mod n is also true.

 2012-09-19, 17:18 #36 alpertron     Aug 2002 Buenos Aires, Argentina 5·269 Posts Thanks.
 2021-03-20, 08:03 #37 Mounir   Mar 2021 1 Posts Hi everyone I know I am a bit late.... BUT I have the answer to your problem: take N = 2^524287 - 1. Notice that 524287 = 2^19 - 1, this is a clue for the proof ;) Indeed, you just need to consider the sequence X(n+1) = 2^X(n) - 1 with X(0) = 19 and proove that all terms of the sequence verify X(n)*X(n+1) divide 2^X(n) - 2. Have a nice day!
 2021-03-21, 16:14 #38 kijinSeija   Mar 2021 1 Posts Hi I would like to know if a^(2^n) mod (2^n-1) = a² when a is between 0 and sqrt(2^n-1) and a is an integer when 2^n-1 is prime and only if it's prime ? I tried this with some Mersenne exposant and it apparently works. It can't be used to eliminate non-Mersenne exponant quickly ? Thanks :D
 2021-03-21, 22:43 #39 MattcAnderson     "Matthew Anderson" Dec 2010 Oregon, USA 23·89 Posts In My Humble Opinion (IMHO) the even numbers are easier to keep track of
2021-03-22, 03:03   #40
LaurV
Romulan Interpreter

Jun 2011
Thailand

222278 Posts

Quote:
 Originally Posted by kijinSeija Hi I would like to know if a^(2^n) mod (2^n-1) = a² when a is between 0 and sqrt(2^n-1) and a is an integer when 2^n-1 is prime and only if it's prime ? I tried this with some Mersenne exposant and it apparently works. It can't be used to eliminate non-Mersenne exponant quickly ? Thanks :D
That is a PRP test in disguise. From Fermat theorem, a^p=a (mod p) if p is prime. Amplify with a, you get a^(p+1)=a^2 (mod p). Replace p with Mp and you have your test. Replace a with 3 and that's what P95 and gpuOwl are doing.

Yes, it can be used to eliminate composite Mp's, but not "fast". Well, not faster than we are doing already.

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