20071017, 02:20  #1 
Dec 2005
2^{2}×23 Posts 
Mersenne theorems question
Are there any theorems that restrict what exponents mersenne prime numbers can have besides the following two?

20071017, 02:40  #2 
May 2003
1547_{10} Posts 
I imagine the second property follows directly from quadratic reciprocity. You could probably make a similar statement using rational quartic reciprocity. I'll try to work it out, and post again.

20071017, 04:16  #3 
May 2003
7·13·17 Posts 
Quartic reciprocity won't immediately apply since q=4p+1 is congruent to 5 mod 8 (when p is an odd prime). So q won't even be a quadratic residue.
Sextic reciprocity shows that if p is a prime, then q=6p+1 divides 2^p1 if and only if p==1 mod 4 and q=A^2+27*B^2 for some integers A and B. This is already pretty messy. (But it does show, for example, that p=5 leads to q=31=2^2+27*1^2, and so we know 312^51.) I imagine that octic reciprocity would give conditions satisfied by the pair p=11 and q=8p+1=89. (One can check that 892^111.) But these conditions keep getting messier (and harder to check). 
20071017, 16:06  #4 
Nov 2003
2^{2}·5·373 Posts 
If p = 3 mod 4, then 2p+1 = 7 mod 8 and hence 2 is a quadratic residue.

20071017, 16:57  #5 
Dec 2005
1011100_{2} Posts 
I am not familiar with quadratic reciprocity and quadratic residues. Would you (plural) elaborate on what they are and why 2 is a quadratic residue?
By the way, it might be helpful if you (plural) know that I am an undergraduate student and so far, I have taken Calculus I, Calculus II, Calculus III and Introduction to Linear Alegbra. 
20071017, 18:11  #6 
May 2003
7×13×17 Posts 
Since you are an undergraduate, I would recommend that if you want to learn about quadratic residues go ahead and take a class on abstract algebra, and then a basic number theory class.
Alternatively, if you are highly motivated, pick up a book on beginning number theory (like Andrew's book "Number Theory" printed by Dover). 
20120405, 06:54  #7 
Apr 2012
7_{8} Posts 
Octic reciprocity
Does anyone actually know the conditions for octic reciprocity? I would really like to know what they are.

20120405, 11:21  #8 
Nov 2003
2^{2}·5·373 Posts 

20120405, 11:45  #9  
"Forget I exist"
Jul 2009
Dumbassville
20300_{8} Posts 
Quote:
x^2=q mod n 

20120405, 15:33  #10  
"Forget I exist"
Jul 2009
Dumbassville
20C0_{16} Posts 
Quote:
Last fiddled with by science_man_88 on 20120405 at 16:04 

20120405, 16:08  #11 
Einyen
Dec 2003
Denmark
5^{2}·127 Posts 
I believe Prime95 uses that factors 2kp+1 are equal to one of these 16 values mod 120:
+1, +7, +17, +23, +31, +41, +47, +49 (mod 120) This is more strict than +1 (mod 8) which leads to 30 values mod 120. 
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