2003-11-30, 05:09 | #1 |
Sep 2003
5×11×47 Posts |
A plot of Log2 (P) vs N for the Mersenne primes
A graph of log_{2}(P) vs N is here:
http://opteron.mersenneforum.org/png/log2_log2_Mn.png Probably we should plot log_{2}(log_{2}(2^{P}-1) vs N instead, but it's the same except for small values of P. More discussion and a similar graph are found at http://www.utm.edu/research/primes/n...tMersenne.html Looking forward to the next imminent data point! Last fiddled with by GP2 on 2003-11-30 at 21:54 |
2003-11-30, 21:53 | #2 |
Sep 2003
2585_{10} Posts |
OK, the plot was modified to show
log_{2}(log_{2}(2^{P}-1) vs N instead of log_{2}(P). Actually, we cheat by plotting log_{2}(P) - 1/(P^{2}-1) instead, which is a close enough approximation. Only the data points for P=2 and P=3 are visibly affected by the second term. We also plot a line with slope 1/e^{gamma} (approximately 0.56145948), where gamma is Euler's constant (0.57721566490153286...). This is the expected slope according to Wagstaff's conjecture, as discussed at http://www.utm.edu/research/primes/n...tMersenne.html Last fiddled with by GP2 on 2003-12-01 at 00:03 |
2003-11-30, 22:00 | #3 |
Sep 2003
5×11×47 Posts |
Also, according to Wagstaff's conjecture, values of P where P=1 mod 4 are more likely to produce a prime than values of P where P=3 mod 4.
Here's a graph of P mod 4 for the Mersenne primes. Sure enough, there are 23 primes with P=1 mod 4, 15 primes with P=3 mod 4, and obviously 2 is in a class by itself, for a total of 39. Interestingly enough, all of GIMPS's primes M35 to M39 have been P=1 mod 4. Will M40 be the same? We'll soon see. Once again, see http://www.utm.edu/research/primes/n...tMersenne.html for more details. |
2003-12-01, 20:24 | #4 |
Sep 2003
5×11×47 Posts |
If we do the calculation, the a priori probability of M39 being prime was 3.474e-06, since 13466917 = 1 mod 4.
If it had been = 3 mod 4, the probability would have been 3.264e-06. As you get to larger and larger exponents, the ratio of probabilities log(6P)/log(2P) tends to go to 1. So the number of P=1 mod 4 and P=3 mod 4 exponents should even out eventually. log 6x / log 2x = (log 6 + log x) / (log 2 + log x) -> 1 as x -> infinity. So there's not much point in prime hunters concentrating only on P=1 mod 4 exponents. At the current leading edge of P = 21.5M, the ratio of probabilities is already down to 1.0625 Last fiddled with by GP2 on 2003-12-01 at 20:28 |
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