[kriesel]

Begin with the content of https://primes.utm.edu/notes/faq/NextMersenne.html

We expect slightly less than 6 Mersenne primes in a 10:1 interval on exponent.
M82589933 is part of an unusual run of (at least) 13 in the 10⁷ to 10⁸ interval.
"For comparison:
1 to 10: 4
10 to 100: 6
100 to 1000: 4
1000 to 10000: 8
10000 to 100000: 6
100000 to 1000000: 5
1000000 to 10000000: 5"
https://www.mersenneforum.org/showpo...70&postcount=5
(The next decade after the above, 10⁷ to 10⁸ is not completely double checked yet.
In 10⁸ to 10⁹ there are relatively few tested so far, with a large amount of further computation to do.)

p for Mp is 1 mod 8 at twice the frequency of 3, 5, or 7 mod 8 up to Mp50.
https://www.mersenneforum.org/showpo...2&postcount=53

The Wagstaff conjecture predicts about 57 Mp below p=10⁹.
https://www.mersenneforum.org/showpo...7&postcount=55

Of the known Mersenne primes, many more are p=1 mod 4 (31) than p=3 mod 4 (19).
https://www.mersenneforum.org/showpo...8&postcount=59
For #51, p= 82589933 = 1 mod 4, so 31 (60.78%) vs 19 (37.25%) 3 mod 4 and 1 (1.96%) of 2 mod 4.
31/19 ~ 1.6316. At exponents > 1000, it's ~2:1.

Mersenne primes are twice as likely to be p=1 mod 4 as 3 mod 4 conjectured, with some empirical tabular support.
Within p=1 (mod 4), Mersenne primes are twice as likely to have p=1 (mod 8) rather than p=5 (mod 8)
https://www.mersenneforum.org/showpo...5&postcount=71

(added following 2021-08-29)
There's naturally a somewhat inverse relationship between
a) which p mod whatever have more found factors, and
b) which p mod whatever have more found Mersenne primes.

I think I may have seen it posted elsewhere, but can't find it now, that exponent p mod 24 shows considerable variation in incidence of known Mersenne primes.

P = 1 mod 24    8    15.69%
P = 2 mod 24    1    1.96%
P = 3 mod 24    1    1.96%
P = 5 mod 24    9    17.65%
P = 7 mod 24    8    15.69%
P = 11 mod 24    3    5.88%
P = 13 mod 24    3    5.88%
P = 17 mod 24    11    21.57%
P = 19 mod 24    5    9.80%
P = 23 mod 24    2    3.92%
total    51

Below p=1000, p mod 24 counts:

p count
1 0
2 1
3 1
5 1
7 4
11 1
13 2
17 3
19 1
23 0

Above p=1000, for known Mersenne primes, p mod 24 is observed with frequency

p count
p n
1 8
5 8
7 4
11 2
13 1
17 8
19 4
23 2

which happens to be a table of counts consisting only of powers of two.

If we break it down further to p mod 120, we get too few known primes per bin to show much.

p mod 120 of the 51 known Mersenne primes
p count
1 3
2 1
3 1
5 1
7 3
13 1
17 4
19 1
29 1
31 1
37 1
(15 excluding 2,3,5 in p mod 120 from 0 to 40)

41 2
43 1
49 2
53 3
61 1
67 1
71 1
77 1
79 1
(13 in p mod 120 from 40 to 80)

83 1
89 3
91 2
97 3
101 3
103 3
107 2
113 2
119 1
(20 in p mod 120 from 80 to 120)

Top of reference tree: https://www.mersenneforum.org/showpo...22&postcount=1