Mp: factors of p-1 and p+1

[paulunderwood]

Has anybody looked at the factors of p-1 and p+1 if prime Mp?

? v=[2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 43112609, 57885161];for(k=1,#v,p=v[k];print(p" "factor(p-1)" "factor(p+1)))

2 matrix(0,2) Mat([3, 1])
3 Mat([2, 1]) Mat([2, 2])
5 Mat([2, 2]) [2, 1; 3, 1]
7 [2, 1; 3, 1] Mat([2, 3])
13 [2, 2; 3, 1] [2, 1; 7, 1]
17 Mat([2, 4]) [2, 1; 3, 2]
19 [2, 1; 3, 2] [2, 2; 5, 1]
31 [2, 1; 3, 1; 5, 1] Mat([2, 5])
61 [2, 2; 3, 1; 5, 1] [2, 1; 31, 1]
89 [2, 3; 11, 1] [2, 1; 3, 2; 5, 1]
107 [2, 1; 53, 1] [2, 2; 3, 3]
127 [2, 1; 3, 2; 7, 1] Mat([2, 7])
521 [2, 3; 5, 1; 13, 1] [2, 1; 3, 2; 29, 1]
607 [2, 1; 3, 1; 101, 1] [2, 5; 19, 1]
1279 [2, 1; 3, 2; 71, 1] [2, 8; 5, 1]
2203 [2, 1; 3, 1; 367, 1] [2, 2; 19, 1; 29, 1]
2281 [2, 3; 3, 1; 5, 1; 19, 1] [2, 1; 7, 1; 163, 1]
3217 [2, 4; 3, 1; 67, 1] [2, 1; 1609, 1]
4253 [2, 2; 1063, 1] [2, 1; 3, 1; 709, 1]
4423 [2, 1; 3, 1; 11, 1; 67, 1] [2, 3; 7, 1; 79, 1]
9689 [2, 3; 7, 1; 173, 1] [2, 1; 3, 1; 5, 1; 17, 1; 19, 1]
9941 [2, 2; 5, 1; 7, 1; 71, 1] [2, 1; 3, 1; 1657, 1]
11213 [2, 2; 2803, 1] [2, 1; 3, 2; 7, 1; 89, 1]
19937 [2, 5; 7, 1; 89, 1] [2, 1; 3, 1; 3323, 1]
21701 [2, 2; 5, 2; 7, 1; 31, 1] [2, 1; 3, 1; 3617, 1]
23209 [2, 3; 3, 1; 967, 1] [2, 1; 5, 1; 11, 1; 211, 1]
44497 [2, 4; 3, 3; 103, 1] [2, 1; 19, 1; 1171, 1]
86243 [2, 1; 13, 1; 31, 1; 107, 1] [2, 2; 3, 1; 7187, 1]
110503 [2, 1; 3, 2; 7, 1; 877, 1] [2, 3; 19, 1; 727, 1]
132049 [2, 4; 3, 2; 7, 1; 131, 1] [2, 1; 5, 2; 19, 1; 139, 1]
216091 [2, 1; 3, 2; 5, 1; 7, 4] [2, 2; 89, 1; 607, 1]
756839 [2, 1; 23, 1; 16453, 1] [2, 3; 3, 1; 5, 1; 7, 1; 17, 1; 53, 1]
859433 [2, 3; 7, 1; 103, 1; 149, 1] [2, 1; 3, 1; 143239, 1]
1257787 [2, 1; 3, 2; 69877, 1] [2, 2; 7, 1; 29, 1; 1549, 1]
1398269 [2, 2; 349567, 1] [2, 1; 3, 1; 5, 1; 127, 1; 367, 1]
2976221 [2, 2; 5, 1; 13, 1; 11447, 1] [2, 1; 3, 1; 401, 1; 1237, 1]
3021377 [2, 6; 17, 1; 2777, 1] [2, 1; 3, 1; 503563, 1]
6972593 [2, 4; 11, 1; 173, 1; 229, 1] [2, 1; 3, 1; 1162099, 1]
13466917 [2, 2; 3, 2; 83, 1; 4507, 1] [2, 1; 149, 1; 45191, 1]
20996011 [2, 1; 3, 4; 5, 1; 7, 2; 23, 2] [2, 2; 83, 1; 63241, 1]
24036583 [2, 1; 3, 1; 4006097, 1] [2, 3; 11, 1; 13, 1; 21011, 1]
25964951 [2, 1; 5, 2; 11, 1; 17, 1; 2777, 1] [2, 3; 3, 1; 13, 1; 83221, 1]
30402457 [2, 3; 3, 1; 7, 1; 37, 1; 67, 1; 73, 1] [2, 1; 23, 1; 660923, 1]
32582657 [2, 10; 47, 1; 677, 1] [2, 1; 3, 1; 5430443, 1]
37156667 [2, 1; 19, 1; 59, 1; 16573, 1] [2, 2; 3, 1; 3096389, 1]
43112609 [2, 5; 7, 1; 11, 1; 17497, 1] [2, 1; 3, 2; 5, 1; 479029, 1]
57885161 [2, 3; 5, 1; 29, 1; 139, 1; 359, 1] [2, 1; 3, 1; 9647527, 1]

p+1=6*q or 12*q , q prime, turns up a lot. If I had spare computing cycles I would concentrate on this type with p-1 divisible by high powers of 2.

[axn]

P+1? No idea.

P-1: http://www.mersenneforum.org/showthread.php?t=5339

[retina]

Originally Posted by paulunderwood

...

p+1=6*q or 12*q , q prime, turns up a lot.

That observation alone doesn't appear to mean anything without some sort of comparative assessment to the p's that don't yield prime Mp's. Is the occurrence of your observation significantly greater than the general pool of tested p's?

And, yes, I know that this sort of numerology is probably all bunkum.

[akruppa]

We pondered this observation before, see axn's link and a few other threads here. The tentative title is "extended crank P minus one smoothness hypothesis", "ECPMOSH", or something like that... I forgot. At any rate, if the smoothness of p-1 has an effect on the probability of Mp being prime, then I'm fairly certain it is due to the smoothness of p-1 affecting the probability that Mp has small prime factors, and thus after trial division, the survivors should be practically equally good candidates for primes again.

[paulunderwood]

c=0;cp=0;p=1;while(p<100000000,p=nextprime(p+1);cp++;if((p%12==11&&isprime((p+1)/12))||(p%12==5&&isprime((p+1)/6)),c++));print(cp" primes less than or equal to "p". With the property are "c". Percentage is "100.*c/cp".")

5761456 primes less than or equal to 100000007. With the property are 249977. Percentage is 4.338

"The property" is "p+1=6*q or 12*q , q prime". But is it significant?

[ewmayer]

Originally Posted by axn

P+1? No idea.

P-1: http://www.mersenneforum.org/showthread.php?t=5339

Post #30 in that thread has a table of p-1 and p+1 factorizations for all but the latest M-prime exponent.

Originally Posted by akruppa

We pondered this observation before, see axn's link and a few other threads here. The tentative title is "extended crank P minus one smoothness hypothesis", "ECPMOSH", or something like that... I forgot.

Ahem ... it's eCPM1SH^TM. Use it. Live it. Embrace it with every fiber of your whole-grain-dosed semicolon, or something.

At any rate, if the smoothness of p-1 has an effect on the probability of Mp being prime, then I'm fairly certain it is due to the smoothness of p-1 affecting the probability that Mp has small prime factors, and thus after trial division, the survivors should be practically equally good candidates for primes again.

Indeed - if someone manages to find a deeper reason for *why* this [alleged] statistical factor-number correlation should occur, we may be able to remove the 'C' from the above trademarked initialism.

[paulunderwood]

Some more speculation; this time about p^2-2.

v=[2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 43112609, 57885161];for(k=1,#v,p=v[k];print(p" "factor(p^2-2)))

2 Mat([2, 1])
3 Mat([7, 1])
5 Mat([23, 1])
7 Mat([47, 1])
13 Mat([167, 1])
17 [7, 1; 41, 1]
19 Mat([359, 1])
31 [7, 1; 137, 1]
61 Mat([3719, 1])
89 Mat([7919, 1])
107 Mat([11447, 1])
127 Mat([16127, 1])
521 [7, 1; 17, 1; 2281, 1]
607 Mat([368447, 1])
1279 [31, 1; 52769, 1]
2203 [23, 1; 79, 1; 2671, 1]
2281 [367, 1; 14177, 1]
3217 [7, 1; 151, 1; 9791, 1]
4253 [7, 2; 369143, 1]
4423 Mat([19562927, 1])
9689 [47, 1; 1063, 1; 1879, 1]
9941 [23, 1; 4296673, 1]
11213 [521, 1; 241327, 1]
19937 [9551, 1; 41617, 1]
21701 [127, 1; 3708137, 1]
23209 [7, 1; 76951097, 1]
44497 [359, 1; 5515273, 1]
86243 [7, 1; 1062550721, 1]
110503 [503, 1; 3391, 1; 7159, 1]
132049 [1223, 1; 14257513, 1]
216091 [109103, 1; 427993, 1]
756839 [153841, 1; 3723359, 1]
859433 Mat([738625081487, 1])
1257787 [1217, 1; 5903, 1; 220217, 1]
1398269 [97, 1; 6703, 1; 3007049, 1]
2976221 [7, 2; 47, 1; 1447, 1; 2658079, 1]
3021377 [23, 1; 167, 1; 2376651647, 1]
6972593 [23, 2; 116959, 1; 785777, 1]
13466917 [47, 1; 3858677733721, 1]
20996011 [233, 1; 1891984883743, 1]
24036583 [7, 1; 17, 1; 4855103548873, 1]
25964951 Mat([674178680432399, 1])
30402457 Mat([924309391636847, 1])
32582657 [257, 1; 1583, 1; 2609509937, 1]
37156667 [73, 1; 2687, 1; 7038546337, 1]
43112609 Mat([1858697054786879, 1])
57885161 [31, 1; 313, 1; 48889, 1; 7063457, 1]

[axn]

Is there any particular reason you left out 42643801?

[paulunderwood]

My source was Wilfrid Keller's list

[axn]

Originally Posted by paulunderwood

My source was Wilfrid Keller's list

Originally Posted by That Page

Last updated March 18, 2009.

The whole page is outdated.

[science_man_88]

Originally Posted by paulunderwood

My source was Wilfrid Keller's list

the list of exponents can be found at:

http://en.wikipedia.org/wiki/Mersenn...ersenne_primes