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Old 2005-08-07, 13:17   #1
robert44444uk's Avatar
Jun 2003
Oxford, UK

192810 Posts
Default Proth Riesel Drag Racing

I have been looking recently at drag racing Proth (k.2^n+1 with fixed
k) series and its corresponding Riesel series k.2^n-1 for the same k,
with a view to finding the k which provides 100 primes for both series
in the lowest series of n.

To do so I have been thankful for Anand Nair's Payamx program, which
sieves k which have no small factors either + or - up to a given
factor. Of course, I might have just checked primorial values of k but
using Anand's program allows greater granularity. I have been managing
the large number of n to check with Maple 9, and checking the final
higher values of n with NewPgen as my sieve and pfgw for the prime

The best result I have achieved so far is for the k most easily
expressed as 3988838823*67#/1858202, for which the 100th prime is
n=41653 for the + series and 20399 for the - series.

There are a number of other interesting challenges I have set myself,
including finding the lowest n for which there are 100 primes + or -,
and the fastest I have achieved is n=4957853627*67#/1858202, with its
100th prime at n=909, although I expect to beat that rather easily as
I explore lower values of n. Good candidates might have 25 primes in
the first 30 n, 42 in the first 100, and 85 by n=500.


Robert Smith
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Old 2005-09-24, 10:35   #2
robert44444uk's Avatar
Jun 2003
Oxford, UK

23×241 Posts
Default A slight improvement

For those who are interested, a slight improvement in my previous posted record:

k=62539727*47#/27791 has its 100th prime for the +series for n= 37907, and for the - series at n=33837


Robert Smith
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