20140502, 21:19  #1 
Jan 2014
2×19 Posts 
Squarefreeness of Mersenne numbers
(That is, primeexponent Mersenne numbers.)
I wonder if we shall hear about developments regarding the mentioned open problem (not necessarily in the near future). In any case, and in a hope to start a discussion about it, here are a few immediate (elementary) results if we assume a square divisor exists: Let p and q be odd primes s.t. . (Not all properties below are unique to our assumption.) 1. It is known thatis a Wieferich prime (moreover, ). 2. From this theorem: (last one implies). 3. , thus for (Recall that if then ). Because , it's also true for. Here are a couple of notes regarding the problem: http://cybrary.uwinnipeg.ca/people/d...e_numbers.html (which I haven't thoroughly looked at as of now) It goes without saying that the facts above won't get us anywhere 
20140502, 22:19  #2  
"Forget I exist"
Jul 2009
Dumbassville
10000011000000_{2} Posts 
Quote:


20140502, 23:51  #3 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
10010010101011_{2} Posts 
Because your statement contains a trivial arithmetic error.
Remove this error and the statement becomes vacuous: "therefore" therefore nothing. was already redundantly stated in (2) and is equivalent to "p is odd". (any odd p, not just ) 
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