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 10

^{7} to 10

^{8} 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, 10

^{7} to 10

^{8 }is not completely double checked yet.

In 10

^{8} to 10

^{9} 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=10

^{9}.

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.

Code:

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:

Code:

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

Code:

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.

Code:

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