20170203, 09:17  #1 
Feb 2017
Belgium
2 Posts 
complement of si
when s0 = 2, s1 = 2*s0*s01, s2= 2*s1*s11 and so on
everybody will recognize lucaslehmer test M5=31 and divides s3 M7=127 and divides s5 lots of people are looking if Mp is prime or not? but what with the complement? s3/M5 = 607 i'm pretty sure that s3/M5, s5/M7, s11/M13, s15/M17, s17/M19, s29/M31 and s59/M61 are prime. in the last case, this means a number with more than 1.000.000.000 numbers (for those numbers i have proof) (for the moment i'am looking for a prime number with just a little over 1.000.000.000 numbers with similar methodology, creating such a number will take several months or even more time. maybe i can find proof more easily than creating the number :) ... in my rare spare time ...) 
20170203, 16:05  #2  
Undefined
"The unspeakable one"
Jun 2006
My evil lair
3^{2}·647 Posts 
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20170203, 16:59  #3  
"Forget I exist"
Jul 2009
Dumbassville
8,369 Posts 
Quote:


20170204, 07:30  #4  
Jun 2003
4,723 Posts 
Quote:
Quote:
It is highly unlikely that you will find any more primes in this sequence. 

20170204, 12:07  #5  
"Forget I exist"
Jul 2009
Dumbassville
8,369 Posts 
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20170204, 13:22  #6  
"Rashid Naimi"
Oct 2015
Remote to Here/There
1,931 Posts 
Quote:
S15 is composite? Or S15/M17 is composite? Thanks in advance. ETA Coming from axn and including S3 (which is not claimed to be prime), logically it should refer to S15/M17. I just wished members here would be a bit more descriptive for comprehension of folks like myself. Last fiddled with by a1call on 20170204 at 13:45 

20170205, 05:14  #7 
Jun 2003
11163_{8} Posts 
I PRP tested s11/M13, s15/M17, and s17/M19, and all three turned out to be composite. I thought it was obvious (from the quoted portion that preceded) that all references were to s3/M5, s5/M7, etc., but I can see now how it cold be confusing. Sorry.
The "these" refer to basic things that people should be doing before making "bold statements". 
20170205, 06:03  #8  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{3}×7×163 Posts 
Quote:
The cofactor of s_{p2}/M_{p} doesn't seem to have any reason to be prime. Sometimes it may be, most of the time it won't. 

20170205, 13:15  #9 
Feb 2017
Belgium
2 Posts 
on the basis, I started with some sequence
f0 = 7 f1 = 7*2702f0 ... fi=fi1*2702fi2 ... the strange thing was that 7 + i*24 seemed to be prime when it was a divisor of fi it worked for the first thousands and thousands and more i and I still don't know where it goes wrong the first time fi mod (7+i24) = 0 seems to hold in most cases (primes of the form 7 + i*24) although I know sometimes you get 0 not on ith place and apparently there are cases when you get 0 and 7 + i*24 is not a prime ... it was to good to be true (like most of the times) 
20170205, 14:05  #10 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
1011001100111_{2} Posts 
An alternative formulation for the LucasLehmer test:
\(s_i=\sqrt2s_{i1}+s_{i2}; s_0=2, s_1=\sqrt2\) \(M_p=2^p1\) is prime iff \(s_{2^{n1}}\equiv0 (mod M_p)\) This is an example of a lucaslehmer sequence which is a generalization of lucas sequences. The normal formula that we know doubles i. \(s_{2i}=s_i^22\) See http://siauliaims.su.lt/pdfai/2009/Ericksen09.pdf for further reading. 
20170215, 02:02  #11  
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
5×31×37 Posts 
Quote:
\(s_i=4s_{i2}s_{i4}; s_0=2, s_2=4\) Looks like I am reinventing the wheel. https://oeis.org/A003500 is already connected to https://oeis.org/A003010 Not surprising really. The standard way of referring to this Lucas sequence of the second kind is \(V_n(4,1)\) It should be possible to work out SNFS polynomials for this sequence. I am unsure whether this will just boil down to x^22 and x^44x^2+2 for n a power of 2. 
