Mersenne prime exponent not randomly distributed?

[Dubslow]

Well, here's the promised data. I use a slightly different definition of "percentile", being (number of primes that did worse)/(total primes excluding current [=total-1]). The data computes the B-powersmooth score and L3 norm score, as opposed to the "standard" B-smooth and L2 norm of the first run.

(Note: The expos in [110503,132049,216091,756839,859433] took 10 seconds each, but all following expos up to M48 completed in all of 5 seconds combined. I'm pretty sure it's in my primality code, which has a cutoff at 1e6 -- traversing a large array apparently takes a while So I guess Python isn't too bad then, still limited by the idiot writing the code )

[Dubslow]

OS X uses big arrays -- that's our conclusion at least. The attached is a standalone version of Ernst's code, for anyone to play with.

Perhaps more amazingly, my Python now runs about as fast -- my primality test was real silly when it came to numbers less than the pre-sieved depth.

time ./mpsmooth.py
<lots of output>
real	0m4.291s
user	0m2.976s
sys	0m0.128s

time ./mpsmooth
<lots of output>
real	0m4.883s
user	0m0.160s
sys	0m0.084s

[ewmayer]

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:

[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:

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.

Serge Batalov comments:

That is indeed what I (re)posted as a riddle (from David Eddy) in the Ooops forum for the users to chew on.

9 heads = 9 times the ratio is below expected geometric mean ratio of 1.48...
Hence the odd slope for the last 10 known primes.

It still doesn't challenge the confidence interval based on all known primes to include the Wagstaff slope.

[ATH]

I tested all 2,760,321 primes between 50M and 100M for the number of factors of p-1 and p+1. There were a total of 14,542,336 factors of the p-1 numbers (5.268 on average) and 14,544,222 (5.269 on average) factors of the p+1 numbers.

Below is the total number of occurences of the different combinations of factors of p-1 and p+1. M49-1 has 8 factors and M49+1 has 3 factors. There are 43,162 primes between 50M and 100M with the 8/3 combination or 1 in every 64 primes. it is far less rare than one might think initially, and if we count primes with 5 or more p-1 factors than p+1 factors the number of them are much larger.

P-1  P+1  # of occurences
2     4   4867
2     5   16015
2     6   23972
2     7   22643
2     8   16462
2     9   9947
2    10   5404
2    11   2708
2    12   1419
2    13   686
2    14   327
2    15   158
2    16   68
2    17   22
2    18   11
2    19   5
2    20   3
2    23   1
3     3   9565
3     4   40373
3     5   74747
3     6   82956
3     7   67237
3     8   43755
3     9   24963
3    10   13234
3    11   6611
3    12   3365
3    13   1577
3    14   698
3    15   391
3    16   169
3    17   80
3    18   43
3    19   12
3    20   3
3    21   3
3    22   1
3    23   1
4     2   4804
4     3   40623
4     4   103991
4     5   139562
4     6   123155
4     7   82603
4     8   48024
4     9   25461
4    10   13060
4    11   6246
4    12   3123
4    13   1466
4    14   710
4    15   342
4    16   161
4    17   82
4    18   36
4    19   14
4    20   13
4    22   4
4    23   1
5     2   16010
5     3   74681
5     4   139260
5     5   147028
5     6   105802
5     7   61117
5     8   31165
5     9   15319
5    10   7391
5    11   3542
5    12   1684
5    13   830
5    14   371
5    15   195
5    16   81
5    17   41
5    18   23
5    19   10
5    20   3
6     2   23786
6     3   83404
6     4   122489
6     5   106493
6     6   64397
6     7   32290
6     8   14998
6     9   6703
6    10   3110
6    11   1482
6    12   663
6    13   329
6    14   141
6    15   75
6    16   23
6    17   24
6    18   9
6    19   2
6    20   1
6    21   1
7     2   22814
7     3   67024
7     4   83443
7     5   60754
7     6   32404
7     7   14269
7     8   6118
7     9   2628
7    10   1112
7    11   533
7    12   225
7    13   117
7    14   54
7    15   21
7    16   14
7    17   4
7    18   4
7    19   1
7    20   1
8     2   16418
8     3   43162
8     4   48213
8     5   31108
8     6   15060
8     7   6057
8     8   2345
8     9   973
8    10   380
8    11   156
8    12   72
8    13   35
8    14   13
8    15   6
8    16   1
8    17   3
9     2   9639
9     3   24843
9     4   25600
9     5   15483
9     6   6871
9     7   2548
9     8   883
9     9   310
9    10   123
9    11   47
9    12   30
9    13   13
9    15   1
9    16   2
9    17   1
10    2   5446
10    3   13193
10    4   12862
10    5   7507
10    6   3045
10    7   1140
10    8   397
10    9   152
10   10   46
10   11   19
10   12   6
10   13   4
10   14   1
10   15   1
10   17   1
11    2   2825
11    3   6691
11    4   6448
11    5   3472
11    6   1440
11    7   504
11    8   161
11    9   55
11   10   16
11   11   12
11   12   2
11   13   2
11   14   1
12    2   1343
12    3   3357
12    4   3054
12    5   1719
12    6   658
12    7   235
12    8   66
12    9   22
12   10   4
12   11   1
12   12   1
13    2   672
13    3   1569
13    4   1426
13    5   819
13    6   297
13    7   114
13    8   40
13    9   18
13   10   6
13   11   1
13   13   1
14    2   303
14    3   772
14    4   681
14    5   385
14    6   147
14    7   51
14    8   14
14    9   7
14   10   3
14   11   1
15    2   150
15    3   400
15    4   321
15    5   212
15    6   72
15    7   28
15    8   9
15    9   2
16    2   88
16    3   179
16    4   144
16    5   80
16    6   33
16    7   8
16    8   4
16   10   1
17    2   20
17    3   82
17    4   83
17    5   41
17    6   18
17    7   5
17    8   1
17    9   1
18    2   16
18    3   33
18    4   30
18    5   13
18    6   6
18    7   4
18    8   2
19    2   13
19    3   16
19    4   23
19    5   7
19    6   6
19    7   1
20    2   4
20    3   6
20    4   4
20    5   4
20    6   2
20    7   1
21    2   1
21    3   5
21    4   6
21    5   3
22    2   1
22    4   1
23    4   2
23    5   1

[ewmayer]

Originally Posted by ATH

I tested all 2,760,321 primes between 50M and 100M for the number of factors of p-1 and p+1. There were a total of 14,542,336 factors of the p-1 numbers (5.268 on average) and 14,544,222 (5.269 on average) factors of the p+1 numbers.

Not sure I understand your point - that only indicates no p+-1 factorization bias for the primes-at-large ... this thread is addressing the question of the possible existence of a statistical bias for M-prime exponents, relative to the primes-at-large.

[ATH]

Sorry I added those averages as an extra information, I was mainly looking for the distribution of primes with various combinations of p-1 and p+1 factors.

I was not trying to do any rigorous statistical analysis like you so feel free to move/remove my posts. I was just trying to get a feel of how common/uncommon/rare these primes are with for example 8 p-1 factors and 3 p+1 factors, and it turns out they are a lot more common than I thought, in that regard these exponents are not very special. Ok, back to your statistical analysis, sorry for the interruption.

[paulunderwood]

74207281 is the p+2 of twin primes. Is there a bias?

[science_man_88]

Originally Posted by paulunderwood

74207281 is the p+2 of twin primes. Is there a bias?

(17:50) gp > y=parapply(r->#parselect(q->q<r && isprime(q+2),MeVec),MeVec);print(y);parforprime(x=1,74207281,if(x==2,z=0);isprime(x+2),l,if(l,z++);if(x==74207281,print(z/primepi(74207281))))
[0, 0, 1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5]
338328/4350601
(17:55) gp > y=parapply(r->#parselect(q->q<r && isprime(q+2),MeVec),MeVec);print(y);parforprime(x=1,74207281,if(x==2,z=0);isprime(x+2),l,if(l,z++);if(x==74207281,print(z*1.0/primepi(74207281))))
[0, 0, 1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5]
0.077765807528660982701010733919290691102
(17:55) gp > 5/44.
%4 = 0.11363636363636363636363636363636363636

this si using the newest parallel version for windows BTW. so in the KNOWN mersenne prime exponents 5 out of 44 are the lower part of a mersenne prime versus about 7% of the primes in general up until that point. by comparison:

(17:55) gp > y=parapply(r->#parselect(q->q<r && isprime(q-2),MeVec),MeVec);print(y);parforprime(x=1,74207281,if(x==2,z=0);isprime(x-2),l,if(l,z++);if(x==74207281,print(z*1.0/primepi(74207281))))
[0, 0, 0, 1, 2, 3, 3, 4, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 12, 12, 12]
0.077765807528660982701010733919290691102
(17:59) gp > 12/44.
%6 = 0.27272727272727272727272727272727272727

okay I may have messed up somewhere as the equal distribution of +/- forms seems counterintuitive to me maybe z wasn't cleared the second time.

[science_man_88]

Originally Posted by science_man_88

okay I may have messed up somewhere as the equal distribution of +/- forms seems counterintuitive to me maybe z wasn't cleared the second time.

okay I realize now I'm just idiotic as if they come in pairs they'd be equal in number

[ewmayer]

[See post #69 in this thread for my analogous analysis re. the 49th discovered M-prime.]

Time for a new entry in this data series for the just-discovered 50th M-prime. Using the classic B-smoothness and the L2-smoothness measures I define in post #32 we have

p = 77232917: p - 1 = 2^2.29.665801. (For comparison, p + 1 = 2.3.7^2.262697).

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

B -smoothness: 77232917 is 823 of 1111, percentile = 26.01
L2-smoothness: 77232917 is 854 of 1111, percentile = 23.22

This is one of the less-(p-1)-smooth M-prime exponents, though by no means extreme in that regard.

[kriesel]

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