20091213, 02:26  #1  
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}·3·641 Posts 
"On factors of Mersenne numbers"  Seiji Tomita
(Am I really the first to post about this item from Thursday's NMBRTHRY?)
Seiji Tomita has posted, on NMBRTHRY, the following: Quote:


20091214, 20:05  #2 
∂^{2}ω=0
Sep 2002
Repรบblica de California
5·2,351 Posts 
I saw the original post on the NT mailing list ... looks to be new, the real question is, is it practically useful? As far at TF goes it seems not, since if e.g. q = 6p+1 or q = 8p+1 is prime it's going to pass the smallfactor sieve and get tested for factorhood, and pretesting to see if the q's are genuine primes is more expensive than seeing if they divide M(p).

20091214, 23:24  #3  
"Bob Silverman"
Nov 2003
North of Boston
2^{2}×1,877 Posts 
Quote:
The idea itself is quite old. The evaluation of the Artin symbol (for reciprocity higher than quadratic) and the ideas behind cubic, quartic reciprocity etc. are well covered in Serre's book: A course on Arithmetic (caveat emptor if you read this book; it is dense). The ideas behind the exact quadratic form to construct for 2kp+1 are covered in Cox's book on quadratic forms and prime representations. The exact quadratic forms given here seem new, but the math behind them is known. 

20091215, 00:31  #4 
Einyen
Dec 2003
Denmark
2·17·101 Posts 
From empirical testing it seems to work the other way as well:
If q=6p+1 divides M_{p}=2^{p}1 then q=7 (mod 8) and q=27x^2+y^2 for some positive x,y. Of the 50,847,534 prime exponents p<10^{9}, this works on all the 1,046,030 exponents where q=6p+1 is a factor (not counting 2^{5}1 as it's own factor q=6*5+1, but it works there too). If q=8p+1 divides M_{p}=2^{p}1 then q = 64x^2+y^2 for some positive x,y. This works on all 773,708 prime exponents p<10^{9} where q=8p+1 is a factor. Last fiddled with by ATH on 20091215 at 00:32 
20091215, 11:20  #5 
Bamboozled!
"๐บ๐๐ท๐ท๐ญ"
May 2003
Down not across
2·3·29·67 Posts 

20091215, 11:48  #6 
"Bob Silverman"
Nov 2003
North of Boston
2^{2}·1,877 Posts 

20091215, 17:45  #7 
"Phil"
Sep 2002
Tracktown, U.S.A.
1120_{10} Posts 
Serre's book doesn't go beyond quadratic reciprocity. I think it is an excellent book to learn from, but is dense, as you say. A good introduction to quadratic reciprocity, padic numbers, quadratic forms, Dirichlet's theorem, and a brief but difficult introduction to modular forms. I reread it a few years ago and was amazed to read the annotations I had written in the book as a student, years earlier, because most of the material did not seem familiar at all to me! Just shows how much you can forget over the years!

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