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 2009-08-20, 02:16 #1 Dougy     Aug 2004 Melbourne, Australia 2308 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 $(p_k)_{k=0}^{\lambda-1}$ such that $p_k=ap_{k-1}+1$ for all $1 \leq k \leq \lambda-1$. 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 co-author. But, in any case, I'd appreciate any feedback. Last fiddled with by Dougy on 2009-08-20 at 02:26
 2009-08-20, 02:59 #2 Kevin     Aug 2002 Ann Arbor, MI 6618 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?
 2009-08-20, 06:11 #3 Dougy     Aug 2004 Melbourne, Australia 23×19 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 2009-08-20 at 06:11
2009-08-20, 10:54   #4
Kevin

Aug 2002
Ann Arbor, MI

433 Posts

Quote:
 Originally Posted by Dougy 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...
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.

 2009-08-24, 12:15 #5 Dougy     Aug 2004 Melbourne, Australia 15210 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.
2009-08-24, 12:20   #6
R.D. Silverman

Nov 2003

26×113 Posts

Quote:
 Originally Posted by Dougy 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.
AFAIK, this particular journal was discontinued some time ago. I did receive
it as an AMS member back in the 80's.

 2009-08-24, 13:44 #7 Dougy     Aug 2004 Melbourne, Australia 23×19 Posts Thanks. It looks like they'll be tricky to track down.
2009-08-24, 13:47   #8
R.D. Silverman

Nov 2003

26×113 Posts

Quote:
 Originally Posted by Dougy Thanks. It looks like they'll be tricky to track down.
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.

2009-08-24, 22:29   #9
Dougy

Aug 2004
Melbourne, Australia

23×19 Posts

Quote:
 Originally Posted by R.D. Silverman 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.
If that's the case, it's probably not worth my time. I'll just say something like "Loh said Sumiyama said..."

 2009-09-05, 22:37 #10 Dougy     Aug 2004 Melbourne, Australia 23·19 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.
 2009-09-11, 21:05 #11 Dougy     Aug 2004 Melbourne, Australia 23·19 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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