On 22 Jun 2001, at 13:42, Gordon Bower wrote:
After seeing a post on this list a few weeks ago I decided to branch
out and try a few ranges from Michael Hartley's page looking for
k*2^n-1 primes. I must say there is a bit of a thrill in actually
discovering a new prime every day I run the program instead of proving
two numbers a month composite. :)
Yes, it does make a change.
Anyway, a few curious observations I made, which surprised me:
I have 2 computers, a P2-350 and P3-500. The program executes nearly 2
1/2 times as fast on the latter as on the former with nothing else
running. Apparently the Proth code takes advantage of a lot of P3
features?
Yes, Proth 6.6 has prefetch code for PIII and Athlon CPUs.
With the same version of prime95 and the same version of proth on each
computer, if you run them both at the same time, under Win2000 proth
gets a higher priority and all the processing power, while under NT4,
it's the other way round, and prime95 has to be stopped or have its
priority reduced in the ini file to not smother proth. Curious. (Why
run them both at once, you ask? If the computer is going to be on all
night anyway, it'd be idle when proth finished a range unless prime95
was ready to take over as soon as proth was done.)
There is a marked difference in the process timeslot allocation
algorithm between NT4 W2K. (IMHO neither is as effective as the
equivalent function in linux 2.2, further improved in linux 2.4, but
that's a different story!) Also between Win95 and Win98. '95 behaves
like NT4, and '98 behaves like W2K. Well, only on uniprocessor
systems, since '9x/ME don't support SMP at all, but I think you get
the drift?
My strategy is:
(1) run Proth at medium priority in factoring only mode to eliminate
candidates with small factors;
(2) on the same system, run PRP at low priority to check the
survivors from stage 1 for probable primes;
(3) on a different system (normally running Prime95), run Proth at
medium priority to verify the probable primes. (If you don't have a
spare system it would be best to do this in a seperate directory so
as to save keep changing the Proth setup!)
(1) takes a lot less time than (2) so even if (2) stops temporarily
that's not a problem. Not much survives (2) so run (3) takes little
time, even though it's much slower per candidate than the others! BTW
so far _every_ probable prime I've found using PRP has been accepted
as a genuine prime by Proth, though this is certainly not guaranteed.
I assumed that one value of k was pretty much the same as any other as
far as execution time and the chance of finding primes. To my surprise
this turned out not to be so: On the P3-500, for most 650k750, it
takes about 5 hours for 16000n32000 and 12 hours for 32000n48000
-- but for k=701 it took less than 2 and just over 6 hours,
respectively. The phenomenon is reproducible, doesn't seem to be an
artifact of other programs or reboots or suchlike. Any number
theorists care to explain what is special about k=701 that makes it
easy to check for primality?
If you break the run down as above you will see that some values of k
yield a much smaller proportion of candidates for psuedo-prime
testing than others. Or, to put it another way, some values of k give
a much higher percentage of k.2^p-1 with small factors than others.
Conversely the slower values of k tend to yield a lot more primes
than the faster ones. In fact, if your trial factoring strategy is
reasonable, your average rate of discovery of primes will not depend
much on the value of k - though it certainly will depend on the
average value of n!
k.2^p+1 behaves similarly. In fact there are some values of k for
which it is _proved_ (mathematically) that there are _no_ k.2^p+1
primes, even though the lowest value of k for which this is true is
still uncertain. (Or at least there was still work in progress last
time I checked.) You may care to look up the Sierpinski Problem if
you're interested in this.
A fun project. Now if Michael would just put a stop to that pesky
server error I could submit a half dozen more primes to him... :)
Yeah, I finished up my raft of blocks a couple of days ago, can't get
any more can't report results. No response to mail messages either.
He may have gone on vacation.
Regards
Brian Beesley
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