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^n-1,
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^n-1 |
|
|
| 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) |
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|
|
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|
** [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^n-1 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^n-1 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^n-1 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
***
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