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-   -   Mp: factors of p-1 and p+1 (https://www.mersenneforum.org/showthread.php?t=17780)

paulunderwood 2013-02-11 11:14

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

[code]
? 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)))
[/code]
[code]
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]
[/code]

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. :smile:

axn 2013-02-11 11:25

P+1? No idea.

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

retina 2013-02-11 11:26

[QUOTE=paulunderwood;328952]...

p+1=6*q or 12*q , q prime, turns up a lot.[/QUOTE]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?

[size=1][color=white]And, yes, I know that this sort of numerology is probably all bunkum.[/color][/size]

akruppa 2013-02-11 11:48

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 2013-02-11 12:06

[code]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".")[/code]

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

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

ewmayer 2013-02-12 00:05

[QUOTE=axn;328954]P+1? No idea.

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

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


[QUOTE=akruppa;328957]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.[/QUOTE]
Ahem ... it's eCPM1SH[sup]TM[/sup]. Use it. Live it. Embrace it with every fiber of your whole-grain-dosed semicolon, or something.

[QUOTE]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.[/QUOTE]
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 2013-02-13 06:47

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

[CODE]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)))[/CODE]

[CODE]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]
[/CODE]

:deadhorse:

axn 2013-02-13 07:34

Is there any particular reason you left out 42643801?

paulunderwood 2013-02-13 07:41

:redface: My source was Wilfrid Keller's [URL="http://www.prothsearch.net/riesel2.html"]list[/URL]

axn 2013-02-13 08:01

[QUOTE=paulunderwood;329274]:redface: My source was Wilfrid Keller's [URL="http://www.prothsearch.net/riesel2.html"]list[/URL][/QUOTE]

[QUOTE="That Page"]Last updated March 18, 2009.[/QUOTE]

The whole page is outdated. :sad:

science_man_88 2013-02-13 20:35

[QUOTE=paulunderwood;329274]:redface: My source was Wilfrid Keller's [URL="http://www.prothsearch.net/riesel2.html"]list[/URL][/QUOTE]

the list of exponents can be found at:

[url]http://en.wikipedia.org/wiki/Mersenne_prime#List_of_known_Mersenne_primes[/url]


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