20090304, 19:41  #12 
Dec 2008
you know...around...
647 Posts 
Oh it's not my CPU time.
I torture the computer at my workplace with that :) 
20090304, 22:44  #13  
Feb 2006
Denmark
2×5×23 Posts 
Quote:
I guess you know Euler found that 41+n*(n+1) is prime for n = 0..39. Jörg Waldvogel and Peter Leikauf found in http://www.sam.math.ethz.ch/~waldvog...clprimes05.pdf that the next occurrence of at least 21 primes is 234505015943235329417+n*(n+1) for n = 0..20. I suspect you have an inefficient program. 

20090305, 19:45  #14  
Dec 2008
you know...around...
647 Posts 
Quote:
Quote:
I wonder why there's not even a sequence for the smallest term for the p series (which would be 2, 5, 13, 19, 43, 43, 73, 73, 109, 109, 245333323, 23181771649 ...). Quote:
For an amateur, I think my program is quite good. You get an idea of how good (or bad) it actually is by looking at the code in the attachment. It's designed for finding at least 12 simultaneous primes from the p+ sequence. And yes, it's very unconventional. Could make it a bit faster, though. But, like I said, it's not my CPU time. 

20090306, 02:36  #15 
Feb 2006
Denmark
230_{10} Posts 
p, length of p+ sequence
4296903559301 12 14334964751111 13 17518107964451 12 30431463129071 14 57725094574571 12 60693437191061 12 1 minute with a C program. Added later: 521999251772081 16 Last fiddled with by Jens K Andersen on 20090306 at 03:22 Reason: Found a length 16 
20090306, 16:59  #16  
Dec 2008
you know...around...
647_{10} Posts 
Quote:
What gives you (and Mr Luhn, Mr Wroblewski, Mr Rosenthal and all the other wellknown prime pattern seekers) the ultimate power boost in searching for these patterns? Any sample program you can put at my disposal? 

20090306, 18:53  #17  
Feb 2006
Denmark
2×5×23 Posts 
Quote:
The details vary but the main things in a search for simultaneous primes are: 1) Use a construction with modular math to only examine cases where none of the numbers have a very small prime factor (for example below 40). 2) Use sieving on an efficient sieve form like arithmetic progressions to quickly eliminate cases where one of the numbers has a relatively small prime factor. 3) Use a fast prp test on the remaining candidates. I usually use the GMP library or PrimeForm/GW for prp testing. 4) Prove the prp solutions. I divide the search space for one of the simultaneous primes into arithmetic progressions of form k*p#+d where d<p#, and p is the largest prime factor to be avoided by the construction. Identify the d values between 0 and p# (or a lower value for an inexhaustive search) for which no candidates of form k*p#+d will have a prime factor <=p in one of the numbers. For each d value, sieve on k to eliminate cases with a small prime factor above p. The sieve often becomes so sparse that I use a form of individual trial factoring after some sieving. If sieving continued then the vast majority of the sieve time would be used on candidates which had already been eliminated. If you want to search for length 17 on a Windows PC then I can prepare and mail an executable in a couple of days when I have better time. My C source is not public and is modified and recompiled by me for each form and length. I don't have a flexible executable where the form can be input. 

20090306, 21:47  #18  
Dec 2008
you know...around...
647 Posts 
So I'm already quite good with the technique I mentioned earlier here:
http://www.mersenneforum.org/showpos...56&postcount=3 Only thing is, I don't know how to effectively sieve an already presieved range for larger factors, since I'm dealing with simultaneous primes, not with separate primes where sieving would be a piece of cake. ... Well, I can think of something, but I'd have yet to test if it is viable. The main obstacle then is the rather slow BASIC which can only handle distinct integers to 2^53 ("long integers" to 2^31). Quote:
At least now I have an idea of where I'm standing in the field of prime number research. Thanks anyway. Last fiddled with by mart_r on 20090306 at 22:11 

20090319, 18:19  #19 
Dec 2008
you know...around...
647 Posts 
So, to make this whole pn(n+1) thingy more interesting:
Code:
p smallest divisor of pn(n+1) 73 17 109 19 2293 29 6163 31 18523 43 42739 47 53509 61 501229 67 11100253 71 17787043 73 38638519 83 40880029 101 65288413 107 432015433 127 2361810469 131 4888301653 137 12276955783 149 96498898819 167 3303436989913 173 3626693315809 179 14456049750829 191 (Also, http://www.primepuzzles.net/conjectures/conj_017.htm  no update since 2005?) 
20090320, 01:52  #20 
Einyen
Dec 2003
Denmark
2×7×223 Posts 
First prime millennia with only 5 primes: 21,185,697,626,___ (primes at 083, 267, 983, 993, 999)
Last fiddled with by ATH on 20090320 at 01:55 
20090320, 07:21  #21 
Dec 2008
you know...around...
647 Posts 

20090320, 15:21  #22  
Nov 2003
2^{2}·5·373 Posts 
Quote:
So? We know there are infinitely many such intervals. What is the point of finding one? May I suggest that those of you participating in this effort might actually participate in a different project? This one has a well defined goal: Assist in the computations that are attempting to disprove the convexity of pi(x). Such a computation would actually establish a THEOREM. Much progress has already been made on narrowing the range in which an exception might be found. I suggest (without trying to flame anyone) that from the viewpoint of mathematical aesthetics, that the efforts under discussion in this forum are more or less pointless? There are several *useful* prime chasing projects that have a realistic goal. Seventeen or Bust is one such. The convexity of pi(x) is another. Another VERY VERY useful computation would be the search for a composite that is both a Lucas PRP *and* an ordinary PRP. Money is offered for this problem by Selfridge, Wagstaff & Pomerance. A good source of problems that mathematicians consider important is given in Richard Guy's book: Unsolved Problems in Number Theory. But, IMO, finding examples of intervals of length 1000 that only contain 5 (or some other fixed number less than 168) serve no useful purpose. We not only know that they exist, but we also know how often an interval of length 1000 will contain 5 primes (on average)... Note: I make the same suggestion to those chasing aliquot sequences in the factoring forum. Unless they have a wellestablished goal, there are better uses for the CPU time. 

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