20181224, 15:00  #1  
Dec 2018
China
11011_{2} Posts 
How did Prof.Cole factor M67?
According to Wiki,
Quote:
But, how could Cole sieve such prime so efficiently? Another question is that, what if the factor appears later? such as the factor of M67 appears like 7618382573*19370772101 I just curious that whether Cole has a efficient TF method so that he could TF M67 in a reasonable time. 

20181224, 15:19  #2  
"Forget I exist"
Jul 2009
Dumbassville
2^{6}·131 Posts 
Quote:
https://primes.utm.edu/notes/proofs/MerDiv.html Last fiddled with by science_man_88 on 20181224 at 15:22 

20181224, 15:23  #3  
Undefined
"The unspeakable one"
Jun 2006
My evil lair
2×3×13×71 Posts 
Quote:


20181224, 15:29  #4 
"Forget I exist"
Jul 2009
Dumbassville
8384_{10} Posts 

20181224, 16:22  #5 
"Robert Gerbicz"
Oct 2005
Hungary
2523_{8} Posts 
Cole has written his way: https://projecteuclid.org/download/p...ams/1183417760
so he used TF "without result", after that basically (not named) the Fermat's method. 
20181224, 16:39  #6  
Feb 2017
Nowhere
5^{2}·131 Posts 
Quote:
If r is a quadratic residue (mod n) [that is, x^2 == r (mod n) is solvable] then r is automatically a quadratic residue (mod p) for each prime factor p of n. With n = 2^67  1, 2 is obviously a quadratic residue (mod n) since 2^67  1 = 2*(2^33)^2 1. Thus, 2 is a quadratic residue modulo every prime factor p of n, so p is congruent to 1 or 7 (mod 8). This condition excludes about half the candidate factors. And, from Cole's 1903 paper to which you have beaten me to providing the link (and which bears the same date as his famous AMS lecture), "ON THE FACTORING OF LAKGE NUMBERS." we have Quote:
BTW, finding this paper involved a search using the parameters cole methods factoring m67 which seemed obvious for the question being asked. This search quickly led me to the paper's title. Doing another search using "on the factoring of large numbers" ams 1903 gave a link that led directly to the above link to the pdf of the paper itself. And, saints be praised, it's not an unsearchable image file! I was able to copypaste text out of it. 

20181224, 16:50  #7 
Dec 2018
China
3^{3} Posts 
Thank you:)
I finally know how he did it. AMAZING! 
20181224, 16:57  #8  
Dec 2018
China
11011_{2} Posts 
Quote:
where 23.53 squals  23 * 53, not 2353/100. quite confusing for the first time to meet such symbol the result is quite the same: Quote:


20181225, 20:35  #9 
"Jeppe"
Jan 2016
Denmark
2^{3}·3·5 Posts 
Wikipedia (which you quote) surely also contains Cole's own paper as a reference, with the DOI link https://doi.org/10.1090%2FS000299041903010799 with which the fulltext PDF can be opened. /JeppeSN

20181225, 21:50  #10 
Sep 2003
101000010011_{2} Posts 
In Cole's article, he mentions that Lucas (in 1876) had found M89 to be composite... erroneously, as it turned out. But it was not proven prime until 1911 by Powers, or eight years after Cole did his work on M67 and wrote his article.
Good thing he chose to spend his "three years of Sundays" working on M67 instead of M89... 
20181226, 06:48  #11 
Dec 2018
China
3^{3} Posts 
Thank you.
Actually I can't believe that such ancient things could appear on the morden Internet. And then I just treat those reference as some historical research 
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