Proth and Riesel Series 100 primes for same k
- 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.