20090820, 02:16  #1 
Aug 2004
Melbourne, Australia
2^{3}×19 Posts 
On prime chains
I've updated the paper I submitted to the arXiv here. It is now entitled "On prime chains." It gives some interesting, but fairly minor results about sequences of primes such that for all .
The second version expands the results of first version, improves the literature review and corrects some typos I made (which would have been very confusing for whomever read the first version). I'm somewhat tempted to submit this to some mediocre journal  but I think i'd prefer it if someone came up with some good ideas, helped make it into a better paper and they could become coauthor. But, in any case, I'd appreciate any feedback. Last fiddled with by Dougy on 20090820 at 02:26 
20090820, 02:59  #2 
Aug 2002
Ann Arbor, MI
433 Posts 
"Lehmer [7] remarked that Dicksonâ€™s Conjecture [3], should it be true, would imply that there are infinitely many prime chains of length \lambda based on the pair (a, b), with the exception of some inappropriate pairs (a, b)."
Out of curiosity, what are the inappropriate pairs? Anything more interesting than just a and b sharing a common factor? 
20090820, 06:11  #3 
Aug 2004
Melbourne, Australia
2^{3}·19 Posts 
There'd also be some others. For example if (a,b)=(3,1) and p(k) is odd, then p(k+1)=3*p(k)+1 is even. Lehmer didn't explain this very well... hmm...
Last fiddled with by Dougy on 20090820 at 06:11 
20090820, 10:54  #4 
Aug 2002
Ann Arbor, MI
433 Posts 
I suppose something similar happens anytime there exists an N for which a=1 mod N and b is coprime to N. Working modulo N, p(k)=p(0)+kb mod N, so you'll always get something divisible by N when k=p(0)*b^1. I feel like there should be a few more ways you can "trivially" guarantee a factor of N in a bounded number of steps if a,b, and N satisfy certain relations, but I'm not prepared to take that on or look up the reference since it's close to 6am local time.

20090824, 12:15  #5 
Aug 2004
Melbourne, Australia
2^{3}·19 Posts 
I'm trying to track down some references from the Loh paper:
Takao Sumiyama, "Cunningham chains of length 8 and 9," Abstracts Amer. Math. Soc., 4 (1983) p. 192. Takao Sumiyama, "The distribution of Cunningham chains," Abstracts Amer. Math. Soc., 4 (1983) p. 489. Has anyone heard of "Abstracts Amer. Math. Soc."? I'm not sure what this means, it could just be a list of talk abstracts or something. Any help would be appreciated. 
20090824, 12:20  #6  
Nov 2003
16444_{8} Posts 
Quote:
it as an AMS member back in the 80's. 

20090824, 13:44  #7 
Aug 2004
Melbourne, Australia
2^{3}×19 Posts 
Thanks. It looks like they'll be tricky to track down.

20090824, 13:47  #8 
Nov 2003
1110100100100_{2} Posts 

20090824, 22:29  #9 
Aug 2004
Melbourne, Australia
98_{16} Posts 

20090905, 22:37  #10 
Aug 2004
Melbourne, Australia
2^{3}·19 Posts 
Here's another "trivial" one that I spotted... if you choose a=1 and b=p_0+p_1. Then the sequence is p_0,p_1,p_0,p_1,... and so on.

20090911, 21:05  #11 
Aug 2004
Melbourne, Australia
2^{3}×19 Posts 
So anyway, I ended up submitting an expanded version of what's on the arXiv. Every paper counts when you're looking for a postdoc.

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