Mersenne prime exponent not randomly distributed?

[ewmayer]

Originally Posted by kriesel

Interesting thread. It seems a bit overdue for an update reflecting p=82589933.

Your wish is my command. :)

Time for a new entry in this data series for the 51st M-prime, discovered in late 2018. Using the classic B-smoothness and the L2-smoothness measures I define in post #32 we have

p = 82589933: p - 1 = 2^2.103.200461. (For comparison, p + 1 = 2.3.7.1966427).

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

B -smoothness: 82589933 is 706 of 1112, percentile = 36.60
L2-smoothness: 82589933 is 814 of 1112, percentile = 26.89

As was the case with its predecessor 77232917, this is one of the less-(p-1)-smooth M-prime exponents.

[kriesel]

Originally Posted by ewmayer

Your wish is my command. :)

It's so much easier to give a gentle nudge than to do it myself. And much more likely to be right that way too. Thanks. So now we have:

https://www.mersenneforum.org/showpo...6&postcount=55 scatter plot (thru M48?)
https://www.mersenneforum.org/showpo...5&postcount=59 summary table of 110k to 57.9M
https://www.mersenneforum.org/showpo...4&postcount=69 M74m
https://www.mersenneforum.org/showpo...2&postcount=76 M77m
https://www.mersenneforum.org/showpo...6&postcount=78 M82.6M

Extended summary table:

p               B -smooth    L2-smooth
--------        ---------    ---------
110503          47.31        56.85
132049          78.57        93.15
216091          99.88        97.62
756839          28.72        17.03
859433          88.34        78.98
1257787         20.32        26.56
1398269          5.42         5.42
2976221         41.08        49.67
3021377         57.65        86.36
6972593         91.83        87.35
13466917        62.51        71.26
20996011        99.75        99.16
24036583         8.03         8.03
25964951        73.05        77.42
30402457        98.74        94.44
32582657        87.41        99.32
37156667        55.44        39.93
42643801        97.02        98.64
43112609        54.13        85.28
57885161        94.04        88.25

was Average:    64.46        68.04

74207281        50.00        84.38
77232917        26.01        23.22
82589933        36.60        26.89

new average:    60.95        65.01

min              5.42         5.42
max             99.88        99.32
# below 50%    8 of 23      8 of 23