Will Eddington's Page

wildrabbitt · 2015-05-17, 11:33

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?

PBMcL · 2015-05-17, 13:58

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.

wildrabbitt · 2015-05-17, 14:03

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.

ATH · 2015-05-17, 14:17

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.

science_man_88 · 2015-05-17, 15:09

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.

edit: https://www.physicsforums.com/thread.../#post-3484943

Reference https://www.physicsforums.com/thread...inus-1.526047/

ewmayer · 2015-05-17, 21:20

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).

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

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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, 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).