mersenneforum.org > Data A plot of Log2 (P) vs N for the Mersenne primes
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 2003-11-30, 05:09 #1 GP2     Sep 2003 5×11×47 Posts A plot of Log2 (P) vs N for the Mersenne primes A graph of log2(P) vs N is here: http://opteron.mersenneforum.org/png/log2_log2_Mn.png Probably we should plot log2(log2(2P-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 GP2     Sep 2003 258510 Posts OK, the plot was modified to show log2(log2(2P-1) vs N instead of log2(P). Actually, we cheat by plotting log2(P) - 1/(P2-1) 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/egamma (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 GP2     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 GP2     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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