20221222, 00:01  #12 
Dec 2022
2^{3}·5·11 Posts 
Yes, that's right. I wonder how you could find that example in reasonable time; I checked it, knowing the exponent, at mersenne.ca, so it's an NP problem for me.
I do think it's normal in mathematics to mean 'prime factors' by the word 'factors'; composites are merely divisors. And that's the way I normally use the two words. The last poster's second statement would imply his first one, but is clearly flawed as (even without this base2 example) it would work for any base, and Sweety has just above provided a counterexample for base 10, and I had indicated how to produce infinitely many  any prime q = kp^2 + 1 has order dividing p (giving this property) in p out of q bases, in a repeating sequence, which reminds one of the similar fact regarding the bases in which any prime has the Wieferich property. The reason I was thinking about this at all is a consideration of the probability of Mersenne numbers being prime. The 1/n rule for factors is the same as for random integers, so a Mersenne with no factors to a certain limit (>> the exponent) should have the same chance of being prime as a random integer with no factors below the same  almost. If the potential factors have an additional constraint to be 1 mod n, their chance of being prime (and therefore, of being a factor) is raised by the factor n/(n1), and so the chance of the whole number being prime changed by the inverse of that, or decreased by 1/n. So I correctly stated in my last thread here that the Eisenstein Mersennes have 2/3 the density (primality chance) of the Gaussian (and ordinary) Mersennes, as their possible factors are all 1 mod 3 (in addition to 1 mod p) while the other classes have no such restriction. The ordinary Mersennes, then, would have a 1/p lower chance assuming the title question true (which I now know it is), but if it were false, this difference would be exactly canceled. 
20221222, 00:53  #13 
Aug 2002
Buenos Aires, Argentina
10111110010_{2} Posts 
From the output of https://www.mersenne.org/report_fact...hi=10000&txt=1 , I've written a 10line program in UBASIC that parses the lines and then prints the lines for which the second number (the prime factor of Mersenne number) modulo the first number (the Mersenne exponent) squared equals 1.
In this case there was no output, so I retried with limits 1000020000, 2000030000, 3000040000 and so on, until the program found the solution I posted with exponent 93077. To make the manual loop faster, I pressed CTRLA, CTRLC to copy all characters from mersenne.org , and then CTRLA, CTRLV in Notepad++ to paste the contents there, deleting the old data. I also deleted manually the header and the footer, so the program in UBASIC only had to parse the comma separated numbers. Notice that this method misses the largest prime factor when the Mersenne number is completely factored. Since I just wanted to find an example and this method included manual operations, it is possible that not all ranges were tested. By the way, I continued the search for exponents up to one million, but I found no more examples. Anyway, most of the prime factors of Mersenne numbers are unknown, so we are missing many possible candidates. Last fiddled with by alpertron on 20221222 at 01:22 
20221222, 15:12  #14  
Feb 2017
Nowhere
2^{4}×3×7×19 Posts 
Quote:
Quote:
There is a wellknown conjecture about the "probability" (loosely speaking) of a Mersenne number with prime exponent being prime. For more details, I suggest you follow the links in this post while you still can. In regard to numbers a^n  1 with composite exponent, the prime factors of the "primitive part" are all congruent to 1 (mod n). See this paper for more details. 

20221222, 23:09  #15 
Dec 2022
2^{3}·5·11 Posts 
I am already acquainted with the mathematical details you mention. I have seen the term 'proper factor' in the sense you mention but the adjective 'proper' is essential to it: if it were normal to use 'factor' alone to mean 'divisor', I think I would have noticed.
The word Mersenne numbers is usually taken as implying a prime exponent, so that composite exponents are not relevant to that. However, they always have algebraic parts that can be prime  which I'll loosely call cyclotomic numbers  and similar calculations can always be performed on them. It seems that the prime probability will always be <= the randominteger value, with equality exactly for Fermat numbers. Possibly the source of some confusion is my reference to the '1/n rule'. This is the rule stating that you can expect a factor of n bits (or n digits in any base) to exist with probability 1/n approximately (it is actually an integral, of course). Deviations from that correspond exactly inversely to deviations in prime probability, as I am using it, since the e^gamma correction factor cancels. alpertron: I didn't think of that; It would have been impossible without a queryable database. And that may well be the only known, as I calculate the number up to Mp to be O(log log p). Thansk for figuring it out, though. 
20221223, 00:37  #16  
Apr 2020
1,021 Posts 
Quote:
Quote:
Quote:
I have never seen "factor" used to refer only to prime factors without that being specified in advance. Quote:
Last fiddled with by charybdis on 20221223 at 00:38 

20221223, 02:35  #17 
Aug 2002
Buenos Aires, Argentina
2·761 Posts 
I extended the search for exponents up to 5 million, but I found nothing.

20221223, 23:19  #18 
Dec 2022
2^{3}·5·11 Posts 
I don't think you need to go any farther, at least with the semimanual method you specified. If you could automatically scan the entire database it might be different.
As to 'factor' vs. 'divisor', all I can say is that I have intuitively understood the distinction I gave for as long as I can remember, and I will continue to use them that way. Even if it's not actually standard, it doesn't _violate_ any standard, and is useful when talking about composite integers. In any case 'divisor' is used commonly when we do want to include everything; we say the sigma function is the 'sum of divisors', never the 'sum of factors'. 
20221224, 00:57  #19  
Apr 2020
3FD_{16} Posts 
Quote:
2. Mathematics isn't going to come crashing down because we have two words for the same concept and some ambiguous terminology. Things aren't always defined in the way that would be most convenient, and most people have learnt to put that to one side and move on. I see that the moderators have given you a chance to do the same. 

20221224, 14:30  #20  
Feb 2017
Nowhere
2^{4}×3×7×19 Posts 
Quote:
If you don't moderate your antisocial behavior yourself and quit crapping up the Forum with your empty egocentrism voluntarily  well, moderators are called moderators for a reason. 

20221226, 18:29  #21 
Aug 2002
Buenos Aires, Argentina
2·761 Posts 
I've written a batch file that includes curl to automate getting factors from Primenet. After 18 hours, I found no other prime of the form p=1 (mod q*q) which is a divisor of 2^{q}  1 for values of the exponent q less than 10^{9}.
Most known prime factors are less than q*q. These factors cannot be congruent to 1 (mod q*q). 
20221227, 01:59  #22 
Dec 2022
2^{3}·5·11 Posts 
Done! That's one very special factor, then, and finding another might be of the same order of difficulty as finding a new Wieferich prime (assuming there is one).
It seems, as this thread has illustrated, that we can all go on as we are accustomed to and that any momentary confusion over 'factor' can be cleared up without hard feelings. I've been interested in math forever and had many discussions about it; I don't remember this ever causing difficulty. However, I probably overstated my case the first time, just out of surprise. 
Thread Tools  
Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
factors of Mersenne numbers  bhelmes  Number Theory Discussion Group  21  20210915 02:07 
Mersenne factors 2*k*p+1  ATH  Miscellaneous Math  7  20201009 11:09 
factors of Mersenne numbers  bhelmes  Miscellaneous Math  8  20200914 17:36 
Distribution of Mersenne Factors  tapion64  Miscellaneous Math  21  20140418 21:02 
Known Mersenne factors  CRGreathouse  Math  5  20130614 11:44 