Problems & Puzzles:
Puzzles
Puzzle 6. Ray Ballinger suggestion
Ray Ballinger notes that for the prime numbers of the form k*2^n+1,
k=12909 is a very productive coefficient since he and Wilfred Keller have detected 73
primes with 73 distinct n values. He notes also that for primes of the form k*2^n1,
k=81555 is the most productive coefficient.
Ray Ballinger suggests to keep tracking this kind of coefficients
(and  of course  the exponents that makes N a prime number! )
Then I offer this page to maintain
these records.


k*2^n+1 


k 
Primes(P) 
n max
[n] 
Index =
P/ln(n) 
Authors 
2863575 
81 
53656
[56729] 
7.437 
Brennen 
12909 
81 
53118
[73000] 
7.444 
Ballinger & Keller 
28995 
90 
28108
[30000] 
8.785 
Keller 


33772


(6/9/1998) 
945561887392230553579269135 



(8/2/03). See below. 


k*2^n1 


k 
Primes(P) 
n max
[n] 
Index =
P/ln(n) 
Authors 
81555 
66 
24351
[28033] 
6.543 
Ballinger & Keller 
22932195 
92 
25038
[27490] 
9.083 
Jack Brennen
(6/9/1998) 





** [n] means limit of known search
***if you want the exponents n, I can sent them by
mail
And, naturally I continue asking for the following
more productive k coefficients.
Solution
Robert Smith wrote (19/11/2002):
I finally cracked, using pfgw, the record for the
k*2^n1 series, after 10 months!
And the k is 147829610027385, which has produced 97
primes in the first 21493 n values, which is, I think 9.7238 on your
measure. I really enjoyed this one. Now I am going to spend a lot shorter
time looking for the + series record.
The choice of k is not too random. It is a result of
searching for the most efficient k values in terms of prime production.
See more at
http://home.btclick.com/rwsmith/pp/payam1.htm
***
The 8/2/2003, Robert Smith wrote:
"... please find below a candidate
(k=945561887392230553579269135)
with 142 primes (k*2^n+1) in less than
110000n.
Thanks to Phil Carmody's fantastic k sieving capability, he was
able to generate in excess of 50,000 Payam number candidates, all of which
are hugely prime up to n=100, for further exploration by me. There was so
much work to do here to eliminate the merely hugely prime series from the
incredibly prime  superlatives fail me here. The side benefit of the work
is that there are about 10 other candidate k which will also break Jack's
record, if you believe in statistical certainties.
I would be grateful if you would credit myself and Phil Carmody
equally for this discovery, along with NewPGen 2.80 for the n sieve
and pfgw for the prime proving."
***
Phil Carmody wrote
(5/5/03):
Recently I've been looking at what I call "Proth Racing", which is
basically what your puzzle 6 is about (you may hae noticed my involvement
with Robert Smith on this puzzle). I've decided to put together a website
about my prime drag racing exploits, which will include some new records.
http://fatphil.org/maths/DragRace/
I've only written a tiny fraction of the pages so far, but there's a
skeleton there already.
Anyway, as a taster for the records that are going to be on those
pages, here are some new records for the k*2^n1 table. The first is the
number which achieves an index of 10 most quickly, and also the largest
number of primes up to n=1000. The second is the fastest number to find
100 primes.
k, P,
n, index
15865502462238176449845, 69,
989, 10.0047931200345
16754719175394037218524715, 100,
5968, 11.5019642903179
Note that the current record for the k*2^n+1 form is equally out of
date, Robert and I have some amazing new numbers in the last few months.
Robert will announce those some time soon.
***
Thomas Ritschel wrote (August
2012)
Quite a while a ago I joined Robert Smith's search for Very Prime
Sequences
(see: http://www.mersenneforum.org/showthread.php?t=9755 ).
I concentrated on the Sierpinski side, e.g. numbers of the the type
k*2^n+1.
After finding a sequence of 172 primes (at n=350000) in 2010
(see: http://www.mersenneforum.org/showpost.php?p=223172&postcount=232)
I was trying to improve this record a little further.
Thanks to an improved preselection software written by Robert Gerbicz I
was able to scan a wide range of k, resulting in a bunch of more than
15000 k's, each yielding 100+ PRPs at n=10000.
Finally, after more than two years, I found a nice pair of record
breakers, yielding 177 and 180 primes up to n=340000:
The first k is 30562993973479965532402725 which produces 180 primes up
to n=338615, P/ln(n)=14.137.
The second k is 32268186233370440249391495 which produces 177 primes up
to n=323172, P/ln(n)=13.952.
And there is another noteworthy k which yields 120 primes until n=10000:
k=29625624785566557571174065 with the 120th prime at n=9805, P/ln(n)=13.057.
Note that Robert Smith also produced quite a few new numbers for the
"minus side", so that the record for the
k*2^n1 form is also out of date.
***
Robert Smith wrote on August 5, 2013:
Please see the link below that announces a k with 204 primes to date. I
found the k, but the workload for prime finding, especially for higher
n, was shared between myself and Thomas Ritschel.
I personally think that beating this candidate will require a lot of
computing power!
http://tech.groups.yahoo.com/group/primeform/message/11407
***
