On prime chains

Dougy · 2009-08-20, 02:16

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.

Kevin · 2009-08-20, 02:59

"Lehmer [7] remarked that Dickson’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?

Dougy · 2009-08-20, 06:11

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...

Kevin · 2009-08-20, 10:54

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.

Dougy · 2009-08-24, 12:15

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.

R.D. Silverman · 2009-08-24, 12:20

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.

Dougy · 2009-08-24, 13:44

Thanks. It looks like they'll be tricky to track down.

R.D. Silverman · 2009-08-24, 13:47

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.

Dougy · 2009-08-24, 22:29

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..."

Dougy · 2009-09-05, 22:37

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.

Dougy · 2009-09-11, 21:05

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.

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, 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)=3p(k)+1 is even. Lehmer didn't explain this very well... hmm... Last fiddled with by Dougy on 2009-08-20 at 06:11*

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2009-08-20, 02:59	#2
Kevin Aug 2002 Ann Arbor, MI 110110001₂ Posts	"Lehmer [7] remarked that Dickson’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?

2009-08-24, 12:15	#5
Dougy Aug 2004 Melbourne, Australia 2³·19 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.

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.

2009-09-05, 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.