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 2015-05-17, 11:33 #1 wildrabbitt   Jul 2014 3×149 Posts Will Eddington's Page Will Eddington's page used to have a proof that a prime divides only one prime-exponent composite Mersenne number. As far as I can tell there's no other proof of this anywhere on the web. His page has gone which is why I'm posting. Am I wrong? Last fiddled with by wildrabbitt on 2015-05-17 at 11:33
2015-05-17, 13:58   #2
PBMcL

Jan 2005

6210 Posts

Quote:
 Originally Posted by wildrabbitt As far as I can tell there's no other proof of this anywhere on the web.
This seems ... unlikely. Huge hint: for a given odd prime q, the order of 2 mod q is unique.

 2015-05-17, 14:03 #3 wildrabbitt   Jul 2014 1BF16 Posts It's a well known theorem. Did you understand what I meant properly? e.g 23 * 89 = 2047 = (2^11) - 1 therefore 23 and 89 do not divide any other prime-exponent composite Mersenne. I've seen the proof, checked it too.
2015-05-17, 14:17   #4
ATH
Einyen

Dec 2003
Denmark

64038 Posts

It is discussed in this old thread: http://www.mersenneforum.org/showthread.php?t=16019

Quote:
 Originally Posted by axn GCD(2^a-1,2^b-1)=2^GCD(a,b)-1. So, for prime exponent mersennes, the answer is no.
The proof for this statement is not given in the thread though.

2015-05-17, 15:09   #5
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

26·131 Posts

Quote:
 Originally Posted by ATH The proof for this statement is not given in the thread though.
re-quoting from above:

Quote:
 GCD(2^a-1,2^b-1)=2^GCD(a,b)-1. So, for prime exponent mersennes, the answer is no.

assume for example that b=2*a; we can show that 2*(2^a-1)+1 = 2^(a+1)-1 and we end up with a general formula of 2^(n+a)-1=2^n*(2^a-1)+(2^n-1) for the 2^n-1 (the part that isn't shown as a multiple of 2^a-1) to have a common factor with (2^a-1) a being the first exponent to have that factor means n must a multiple of a but if n is a multiple of a then n+a is also a multiple of a so only if the exponent b is a multiple of a will it have that factor in common since the primes are all relatively coprime it is shown that no ratio of the exponents fit this and the proof is done.

Last fiddled with by science_man_88 on 2015-05-17 at 15:24

 2015-05-17, 21:20 #6 ewmayer ∂2ω=0     Sep 2002 República de California 5·2,347 Posts Is Will Eddington any relation to the late great astrophysicist Arthur, who famously organized the 1919 British solar-eclipse expedition which confirmed a key cornerstone of Einstein's general relativity theory? Yes, I *know* it's not really a double-D (he said to the buxom lass).

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