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2016-01-20, 01:50   #69
ewmayer
2ω=0

Sep 2002
República de California

3·53·31 Posts
M(74207281)

Time for a new entry in this data series for the just-discovered 49th M-prime. Using the classic B-smoothness and the L2-smoothness measures I define in post #32 we have

p = 74207281: p - 1 = 2^4.3.5.7.44171. (For comparison, p + 1 = 2.107.346763).

Compared to a sample of over 1000 of its peers (primes in [p-10^4, p+10^4]; cf. attachment #2),

B -smoothness: 74207281 is 574 of 1146, percentile = 50.00
L2-smoothness: 74207281 is 180 of 1146, percentile = 84.38

Since B-smoothness only cares about the largest prime factor - here 44171 is ~60% the size of 74207280, logarithmically speaking - we land smack in the middle of the sample according to that metric. L2-smoothness includes all the factors, so there the small factors boost the resulting percentile.

-------------------

Further: With a view towards large-exponent asymptotics I did various best-fit experiments, starting with the full 49-point 'knowns' dataset and truncating various chunks at the low end. Here x is the index of the M(p) (in size-sorted order rather than by discovery date, obviously) and y = log2(p). As my comment notes, the final sample of just the largest 9 M(p)s leads to a big shift in the fit-line:
Code:
[1]	Least-squares of full 49-point dataset gives slope =   0.5465, y-intercept =   1.1208

[2]	Omitting 10 smallest M(p): Sample size = 39, xavg =  30.0000, yavg =  17.6238
Least-squares omitting 10 smallest M(p) gives slope =   0.5252, y-intercept =   1.8685

[3]	Omitting 20 smallest M(p): Sample size = 29, xavg =  35.0000, yavg =  20.2390
Least-squares omitting 20 smallest M(p) gives slope =   0.5240, y-intercept =   1.8977

[4]	Omitting 30 smallest M(p): Sample size = 19, xavg =  40.0000, yavg =  23.0612
Least-squares omitting 30 smallest M(p) gives slope =   0.4351, y-intercept =   5.6575

[5]	Omitting 40 smallest M(p): Sample size = 9, xavg =  45.0000, yavg =  25.1945
Least-squares omitting 40 smallest M(p) gives slope =   0.1895, y-intercept =  16.6660	<*** Holy crap! ***
Using the 5 distinct regressions to predict both the 49th and the 50th M-prime we get a wide range of estimates:
Code:
E.g. using bc -l:
l2 = l(2)
a = 0.1895; b = 16.6660
x=49;lgp=a*x+b;e(lgp*l2)
x=50;lgp=a*x+b;e(lgp*l2)

[1] p49 ~= 250337642;	p50 ~= 365627666
[2] p49 ~= 203901903;	p50 ~= 293441972
[3] p49 ~= 199761040;	p50 ~= 287243698
[4] p49 ~= 132131573;	p50 ~= 178642487
[5] p49 ~=  64890322;	p50 ~=  73998875
Attachment #1 has the above in graphical form - clearly, omitting the smallest 40 M(p) gives much too small a statistical sample to take seriously in the viewed-at-large sense, but it is rather striking how the most recent 10 M(p) line up quite neatly on a very different trendline.