20050714, 16:30  #1  
"Bob Silverman"
Nov 2003
North of Boston
2^{3}×3×311 Posts 
Quote:
"Knowing if M_p 2 is also prime" is somewhat interesting. However, I would like to observe that in the current stateoftheart in factoring, that trying to factor M_p 2 is HOPELESS, except for the smallest p. The level of effort spent so far has not been unreasonable. (IMO). However, I think spending yet more time is unlikely to lead to success(es). Allow me to instead suggest an alternative project which does have some hope of success: Extending the Cunningham project to 'homogeneous form', i.e. numbers of the form A^n  B^n with (A,B) = 1, B>1. [Cunningham is just B = 1] I have already done some modest work in this area. I have completed the following: ("Base x to y" means that I have completed A^n  B^n for A = x and all B < A up to exponent y with (A,B) = 1) Base 3 to 330 Base 4 to 256 Base 5 to 225 Base 6 to 195 Base 7 to 165 Base 8 to 155 Base 9 to 135 Base 10 to 130 Base 11 to 130 Base 12 to 120 I will make these results available to anyone who asks. I don't post them here; the current tables are ~600Kbytes total. Perhaps some of you might like to extend these? Such an effort would be achievable. I am also cross posting this to the 'factoring' sub group. 

20050714, 20:17  #2 
Aug 2002
Buenos Aires, Argentina
2·727 Posts 
None of the Computational Number Theory projects can really terminate. Even for the projects where 100 to 200 new results are found each year (as is the case for the Cunningham Tables), as soon there are only a few numbers to finish factoring the tables are extended, so the process can continue indefinitely.
The same occurs with factoring x^y+y^x, partition numbers and others. When the number of candidates is small, it is surely impossible to finish the task for the remaining candidates, as is the case of Seventeen or Bust project even when they are using about one thousand computers. Well, I decided to find prime factors of M(p)2 and prime factors of numbers near googolplex (in my site). I even have a page about factors near googolplexplex. Of course they will never terminate but I like them. 
20050714, 20:35  #3  
Aug 2002
Buenos Aires, Argentina
10110101110_{2} Posts 
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20050714, 20:41  #4  
"Bob Silverman"
Nov 2003
North of Boston
2^{3}·3·311 Posts 
Quote:
(1) Your discussion is irrelevant to what I said. It is clear that current projects can be extended arbitrarily. What I was discussing is doing something *achievable*, as opposed to doing something that is *hopeless*. Factoring M_p 2 for the larger p has ZERO hope of presently succeeding. (2) You wrote: "When the number of candidates is small, it is surely impossible to finish the task for the remaining candidates, as is the case of Seventeen or Bust project even when they are using about one thousand computers." This makes no sense at all. It is gibberish. Why does a small number of candidates imply that finishing the remaining candidates is impossible?? This is clearly not true for Seventeen or Bust. 

20050714, 21:01  #5  
Aug 2002
Buenos Aires, Argentina
2×727 Posts 
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Sorting the exponents in ascending order we have: 698207 995972 1013803 1157446 1337287 5054502 7830457 9167433 we can extrapolate it in order to find that they will never find the nine primes remaining with current knowledge. 

20050715, 12:56  #6  
"Bob Silverman"
Nov 2003
North of Boston
2^{3}·3·311 Posts 
Quote:
and can not figure out what is being said. "Easier first half" First half of WHAT? "Second half" Again, half of WHAT? "The exponents will be very high, especially for the last prime" If you are discussing the expected value of the smallest exponent which yields a prime for each of the remaining Sierpinski candidates, then you have absolutely NO basis for claiming they will be "very high". Firstly because "very high" is meaningless, and secondly because we do not know how primes of the form k*2^n + 1 are asymptotically distributed. We have no way of guessing what will be the smallest value of n for each k. *I* don't know enough to make such an extrapolation. So how can you possibly assert "never find the nine primes remaining"??? If we assume that for a given k, that k*2^n+1 is prime with probability C(k)/n [which is what PNT implies] where C(k) is a constant depending on k, then we can estimate n such that the probability there exists a j < n with k*2^j+1 being prime is greater then 1/2. *I* will be very surprised if we need to search above exponent = 10^8 to find the remaining primes. Such a search effort is very large, but doable. 

20050715, 13:27  #7  
Aug 2002
Buenos Aires, Argentina
2·727 Posts 
Quote:
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Look for example at: http://www.prothsearch.net/sierp.html You have 40 fivedigit exponents listed there. You have 10 sixdigit exponents listed there plus 2 in SB, totalling 12. There are 6 sevendigit exponents found in SB. There are 9 unknown exponents. Extrapolating it appears that the largest exponent will have at least 10 digits. Of course it is not proved that all remaining sequences have a prime k*2^n+1. In the case that one of them is a Sierpinski sequence, the search time will be infinite. 

20050715, 18:34  #8 
"Phil"
Sep 2002
Tracktown, U.S.A.
3·373 Posts 
Note from the moderator:
Since the original Twin Primes thread was being used as a coordination thread for an ongoing project, I have split off the recent postings which seemed somewhat offtopic and used them to start a new thread. If anyone objects, or would like a different heading, please contact me by PM. Phil 
20050715, 21:56  #9  
"William"
May 2003
New Haven
2^{2}·593 Posts 
Quote:
It's slightly more complicated because the C(k) values  Proth Weights that are easily estimated with Jack Brennan's applet  are different, so the probability of having found a prime by "n" is different for each k. Some of the remaining numbers, especially 67607, have very low weight. It's likely that we will need to go much higher than 10^8 to find all the remaining primes. The hardest part of this is figuring out how to present the results to people with poor probability intuition. There are some standard probability "paradoxes" here, such as increasing residual life, that can cause people to tune out the whole heuristic. The approach I found most successful in communicating these results was to calculate, for each "j", the range of values where the most probable number of primes found is "j." The range for all 17 primes doesn't start until 1.8 x 10^{13}. The original doorway post is on this thread (although software changes have garbaged the original embedded html nonbreaking spaces) http://www.freedc.org/forum/showthr...&threadid=2268 vjs has posted updates listing the primes next to heuristic. Here's a recent one. http://www.freedc.org/forum/showthr...hlight=108G18T The sixth and seventh primes were "right on schedule," and the eighth is only "slightly early." William Last fiddled with by wblipp on 20050715 at 22:04 

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