mersenneforum.org > Math Will prime finding become easier?
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 2007-12-22, 05:34 #1 jasong     "Jason Goatcher" Mar 2005 3·7·167 Posts Will prime finding become easier? I have a hypothetical question. Let's say that computer power suddenly decided it wanted to double exactly every 2 years and it increased along the smoothest(visually) possible curve. Let's say that algorithmic improvements have gone bye-bye, but prime-testing abilities have increased along the same line as computer power, except it goes by the rule that a 50% bigger test takes twice as long, which is what I've been told for LLR. Now assuming the average prime-searcher attempts to keep the challenge the same(the average n-value tested would increase 1.5^x as computer power would increase by 2^x) would primes discovered in these ever increasing ranges tend to increase or decrease? Statistically, would the discovery of primes become more or less common? I'm hoping someone else can express it better, I don't have the chops to do it properly. :(
 2007-12-22, 09:25 #2 davieddy     "Lucan" Dec 2006 England 2×3×13×83 Posts For good heuristic reasons, we expect 1.78 Mersenne primes between exponents 2^x and 2^(x+1). The number of prime exponents to test doubles, and the flops per exponent for an LL test quadruples. So we can expect 1.78 Mersenne primes each time GIMPS flops/s increases by a factor of 8. This suggests that the rate of finding Mersennes should become constant, given Moore's Law for GIMPS throughput. I reckon we are approaching this steady state from the "easy" side, in other words, finding primes will get harder but the rate of discovery will level out. Last fiddled with by davieddy on 2007-12-22 at 09:36
 2007-12-22, 10:07 #3 davieddy     "Lucan" Dec 2006 England 2×3×13×83 Posts http://mersenne.org/ips/stats.html You will see that the exponential increase in GIMPS flops/s "hit the wall" in 2004. Try examining my proposition that we get 1.78 primes each time flops/s increases by a factor of 8 up till then. BTW it's about time this page was updated methinks. PS the last five primes occurred in a freakish burst, although 4 or more between successive powers of two was not eactly unexpected statistically speaking. Last fiddled with by davieddy on 2007-12-22 at 10:14
 2007-12-22, 21:17 #4 jasong     "Jason Goatcher" Mar 2005 3×7×167 Posts Okay, new question, kind of the same thing, but not quite. We're dealing with k*2^n plus or minus 1, fixed n, although input on fixed k would be very much appreciated. Okay, for fixed n, correct me here, when you get above a certain threshold of k, time for a test increases a lot, but remains steady for a good long while(yes I know this isn't mathematically sound, and I apologize for that). Assuming the threshold in the previous paragraph is reached(I just realized the stuff added a few words later makes this first part look stupid, Sorry) I add the following: You obtain the n-weights on the standard prime-finding equation for k=1 to 100,000 n=1 to 500,000 and p=1 to 100,000. So, my question is, all things being equal(yes, I know they're not), wouldn't the lowest weight ns be the best bet for finding primes, twin primes, etc., since using the equation in the above way means all tests stay the same length? There's a lot of stuff I've left out, but I'd rather correct confusion than add stuff that makes confusion worse. Mind you, I'm no expert on this, but I tend to hang with very creative people, and I have a pretty good nose for BS. This concept stuck out for me, the idea of sieving large amounts of fixed n values. I'm sure other people have come up with this idea, but Primegrid is searching n=333,333 for twin primes, and I think they should be searching n=344,208. When I made my comment, they said when they gave up on n=333,333, they would move on to n=500,000, another pretty n. This frustrates me to no end. I don't necessarily need the n-value to specifically be n=344,208, but could someone please try to reason with them? Last fiddled with by jasong on 2007-12-22 at 21:29
 2007-12-22, 21:25 #5 retina Undefined     "The unspeakable one" Jun 2006 My evil lair 137748 Posts What is your reasoning for 344,208 being better than 333,333? What do you base your arguments on? Last fiddled with by retina on 2007-12-22 at 21:26
2007-12-25, 05:08   #6
jasong

"Jason Goatcher"
Mar 2005

3×7×167 Posts

Quote:
 Originally Posted by retina What is your reasoning for 344,208 being better than 333,333? What do you base your arguments on?
I'm almost afraid to answer this question.

To be perfectly honest, it's a bit like hero worship. If you had a topic that fascinated you, you weren't even close to being an expert, but knew someone who seemed to have almost god-like powers in that area, and they gave you advice on that subject, would you want a second opinion? My friend says n=344,208 is a tremendously better number, he's been doing a lot of research on this stuff, and he is(or was) a major player in the prime-finding scene.

I've gotten in trouble for mentioning him in the past,(waiting for him to publish something before I can unloose my tongue) but he was fairly enthusiastic about 344,208, although I suspect it had a lot to do with the fact that it was close to 344,208. There may be better choices between that and n=400,000.

I've probably totally lost credibility by now, but if someone could remind me of the name of a program that tells the weights of various ks or ns, and maybe how to make a looping program that covers ns from about 100,000 to 500,000 and sticks them in a file, I will run it and post the results, good or bad. It would also be great if the results could be in three columns: n-value, +1 weight, -1 weight. Or if someone knew the name of the program I could google how to do looping.

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