On prime chains
I've updated the paper I submitted to the arXiv [URL="http://arxiv.org/abs/0908.2166"]here[/URL]. It is now entitled "On prime chains." It gives some interesting, but fairly minor results about sequences of primes [tex](p_k)_{k=0}^{\lambda1}[/tex] such that [tex]p_k=ap_{k1}+1[/tex] for all [tex]1 \leq k \leq \lambda1[/tex].
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. 
"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? 
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...

[QUOTE=Dougy;186658]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...[/QUOTE]
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. 
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. 
[QUOTE=Dougy;187260]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.[/QUOTE] AFAIK, this particular journal was discontinued some time ago. I did receive it as an AMS member back in the 80's. 
Thanks. It looks like they'll be tricky to track down.

[QUOTE=Dougy;187271]Thanks. It looks like they'll be tricky to track down.[/QUOTE]
Be aware that it was not peer reviewed in any way. Any member could submit an abstract at any time. Said abstract did not need to make sense. 
[QUOTE=R.D. Silverman;187272]Be aware that it was not peer reviewed in any way. Any member
could submit an abstract at any time. Said abstract did not need to make sense.[/QUOTE] If that's the case, it's probably not worth my time. I'll just say something like "Loh said Sumiyama said..." 
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.

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.

All times are UTC. The time now is 14:29. 
Powered by vBulletin® Version 3.8.11
Copyright ©2000  2020, Jelsoft Enterprises Ltd.