mersenneforum.org 2^(2^127-1)- 1, what's low boundary by trial factor?
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 2022-02-04, 06:49 #1 Shen   Jan 2017 5 Posts 2^(2^127-1)- 1, what's low boundary by trial factor? I came to know that such number has been checked there is no prime factor below 5*10^51 by https://primes.utm.edu/mersenne/ . What's the latest result about the low boundary ? what's the suitable program to check it (by gpu or cpu)? I am coding a new method based on probability for factor(2^p- 1)，quite different with other famous algorithms. Thus I want to attack the cantor's conjecture. Thanks for your valuable information.
2022-02-04, 11:13   #2
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

23×431 Posts

Quote:
 Originally Posted by Shen I came to know that such number has been checked there is no prime factor below 5*10^51 by https://primes.utm.edu/mersenne/ . What's the latest result about the low boundary ? what's the suitable program to check it (by gpu or cpu)? I am coding a new method based on probability for factor(2^p- 1)，quite different with other famous algorithms. Thus I want to attack the cantor's conjecture. Thanks for your valuable information.
The current limit of the trial division for M(M(127)) is 145P*M127 (1.45*10^17*M127, about 2.467*10^55), see http://www.doublemersennes.org/mm127.php

Prove or disprove the primality of M(M(127)) is very interesting, since it will disprove either Catalan–Mersenne number conjecture or the New Mersenne conjecture, if M(M(127)) is proved prime, then the New Mersenne conjecture would be disproved, since (2^M(127)+1)/3 is divisible by 886407410000361345663448535540258622490179142922169401, hence composite, and thus M(127) will be a counterexample of the New Mersenne conjecture, on the other hand, if M(M(127)) is proved composite, then the Catalan–Mersenne number conjecture would be disproved, since M(M(127)) is the next number after M(127) in the Catalan–Mersenne sequence, and since if c is composite then 2^c-1 is also composite, M(127) will be the largest prime in the Catalan–Mersenne sequence.

Last fiddled with by sweety439 on 2022-02-04 at 11:19

 2022-02-04, 11:18 #3 ATH Einyen     Dec 2003 Denmark 22×72×17 Posts The limit is k=145P. So the factor limit is: 2*k*p+1 = 2 * 145P * (2127-1) + 1 ~ 4.93*1055 Last fiddled with by ATH on 2022-02-04 at 11:19
2022-02-04, 11:58   #4
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

23×431 Posts

Quote:
 Originally Posted by ATH So the factor limit is: 2*k*p+1 = 2 * 145P * (2127-1) + 1 ~ 4.93*1055
The factor limit should be k*p+1

 2022-02-04, 12:20 #5 Shen   Jan 2017 5 Posts Thank you! sweety439,ATH. according to http://www.doublemersennes.org/math.php, it seems that the limit should be 2*145P *(2^127-1) + 1. Last fiddled with by Shen on 2022-02-04 at 12:45 Reason: add information
2022-02-04, 17:12   #6
bur

Aug 2020
79*6581e-4;3*2539e-3

22·7·19 Posts

Quote:
 Originally Posted by sweety439 Prove or disprove the primality of M(M(127)) is very interesting, since it will disprove either Catalan–Mersenne number conjecture or the New Mersenne conjecture
Indeed interesting. But since Catalan only conjectured that they are "prime up to a certain limit" the first five might well be the only ones. So it might seem more probable that it is composite? Well, for almost all numbers this is true though. Maybe what I mean is rather, it would be less of a surprise if Catalan-Mersenne ended after 5 terms - which technically wouldn't even disprove it - than the New Mersenne conjecture being wrong?

I remember having read here once that in a series of double exponential numbers there is often only a limited number of primes. Where does this come from?

Last fiddled with by bur on 2022-02-04 at 17:17

2022-02-04, 19:12   #7
charybdis

Apr 2020

2EC16 Posts

Quote:
 Originally Posted by bur I remember having read here once that in a series of double exponential numbers there is often only a limited number of primes. Where does this come from?
The Prime Number Theorem says that the probability that n is prime tends to 1/log(n) as n tends to infinity. If you have a sequence that grows doubly exponentially, then the denominators log(n) grow exponentially. The expected number of primes in the sequence will then be the sum of a geometric series, which converges. Hence we expect finitely many primes, though for most of these series there's no hope of being able to prove finiteness.

[Note: for some doubly exponential sequences, such as Fermat numbers, the lack of small factors means that the 1/log(n) is instead loglog(n)/log(n). This doesn't affect the convergence of the series.]

The Catalan-Mersenne sequence grows far faster than doubly exponentially. It almost certainly contains no primes other than the ones known.

2022-02-04, 21:25   #8
kriesel

"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest

2×72×67 Posts

Quote:
 Originally Posted by sweety439 The factor limit should be k*p+1
Not sure why you drop the 2 there.
M(11)%23=0, k=1. Any factors of Mersenne numbers with prime exponents are of the form f = 2 k p + 1, where p is the prime exponent, and k is a natural number. https://primes.utm.edu/notes/proofs/MerDiv.html

From the end of my GTX1650 GPU mmff run to k=145P for MM127:
Code:
[Sep 10 21:38] M127        [185-186]:  99.6% 4596/4620,956/960 |   n.a.  | 554.12s | 36m56s | 191.52G | 345.63M/s | 1050165 |   n.a.%  | kriesel@emu/gtx1650
[Sep 10 21:47] M127        [185-186]:  99.7% 4597/4620,957/960 |   n.a.  | 554.00s | 27m42s | 191.52G | 345.70M/s | 1050165 |   n.a.%  | kriesel@emu/gtx1650
[Sep 10 21:56] M127        [185-186]:  99.8% 4600/4620,958/960 |   n.a.  | 554.16s | 18m28s | 191.52G | 345.60M/s | 1050165 |   n.a.%  | kriesel@emu/gtx1650
[Sep 10 22:05] M127        [185-186]:  99.9% 4605/4620,959/960 |   n.a.  | 554.13s |  9m14s | 191.52G | 345.62M/s | 1050165 |   n.a.%  | kriesel@emu/gtx1650
[Sep 10 22:15] M127        [185-186]: 100.0% 4617/4620,960/960 |   n.a.  | 554.23s |  0m00s | 191.52G | 345.56M/s | 1050165 |   n.a.%  | kriesel@emu/gtx1650
no factor for MM127 in k range: 144115188075855872 to 145000000000000000 (186-bit factors) [mmff 0.28 mfaktc_barrett188_M127gs]
Here M127=p = 127 bits
kmax = 145E15 ~ 57.0088 bits
f = 2 k p + 1 ~ 1 + 127 + 57+delta ~185+delta bits which mmff calls 186 bits, because that's what it takes to store or manipulate 185.xxxx bit of data.

MMFF 0.28 currently supports TF up to 188 bits for MM127. One of these years I'll get back to it.
https://mersenneforum.org/showpost.p...&postcount=382

2022-02-05, 10:10   #9
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

23×431 Posts

Quote:
 Originally Posted by kriesel Not sure why you drop the 2 there.
Well, see https://oeis.org/A001917, for any n, all prime factors of Phi(n,2)/gcd(Phi(n,2),n) (where Phi is the cyclotomic polynomial, if n is prime, then this number is the Mersenne number 2^n-1, and if n is power of 2, then this number is the Fermat number 2^(n/2)+1) (they are the primes p such that znorder(Mod(2,p)) = n) are of the form k*n+1

Code:
p   znorder(Mod(2,p))   k=(p-1)/znorder(Mod(2,p))
3,2,1
5,4,1
7,3,2
11,10,1
13,12,1
17,8,2
19,18,1
23,11,2
29,28,1
31,5,6
37,36,1
41,20,2
43,14,3
47,23,2
53,52,1
59,58,1
61,60,1
67,66,1
71,35,2
73,9,8
79,39,2
83,82,1
89,11,8
97,48,2
101,100,1
103,51,2
107,106,1
109,36,3
113,28,4
127,7,18
131,130,1
137,68,2
139,138,1
149,148,1
151,15,10
157,52,3
163,162,1
167,83,2
173,172,1
179,178,1
181,180,1
191,95,2
193,96,2
197,196,1
199,99,2
211,210,1
223,37,6
227,226,1
229,76,3
233,29,8
239,119,2
241,24,10
251,50,5
257,16,16
263,131,2
269,268,1
271,135,2
277,92,3
281,70,4
283,94,3
293,292,1
307,102,3
311,155,2
313,156,2
317,316,1
331,30,11
337,21,16
347,346,1
349,348,1
353,88,4
359,179,2
367,183,2
373,372,1
379,378,1
383,191,2
389,388,1
397,44,9
401,200,2
409,204,2
419,418,1
421,420,1
431,43,10
433,72,6
439,73,6
443,442,1
449,224,2
457,76,6
461,460,1
463,231,2
467,466,1
479,239,2
487,243,2
491,490,1
499,166,3
503,251,2
509,508,1
521,260,2
523,522,1
541,540,1
547,546,1
557,556,1
563,562,1
569,284,2
571,114,5
577,144,4
587,586,1
593,148,4
599,299,2
601,25,24
607,303,2
613,612,1
617,154,4
619,618,1
631,45,14
641,64,10
643,214,3
647,323,2
653,652,1
659,658,1
661,660,1
673,48,14
677,676,1
683,22,31
691,230,3
701,700,1
709,708,1
719,359,2
727,121,6
733,244,3
739,246,3
743,371,2
751,375,2
757,756,1
761,380,2
769,384,2
773,772,1
787,786,1
797,796,1
809,404,2
811,270,3
821,820,1
823,411,2
827,826,1
829,828,1
839,419,2
853,852,1
857,428,2
859,858,1
863,431,2
877,876,1
881,55,16
883,882,1
887,443,2
907,906,1
911,91,10
919,153,6
929,464,2
937,117,8
941,940,1
947,946,1
953,68,14
967,483,2
971,194,5
977,488,2
983,491,2
991,495,2
997,332,3
1009,504,2
1013,92,11
1019,1018,1
1021,340,3
1031,515,2
1033,258,4
1039,519,2
1049,262,4
1051,350,3
1061,1060,1
1063,531,2
1069,356,3
1087,543,2
1091,1090,1
1093,364,3
1097,274,4
1103,29,38
1109,1108,1
1117,1116,1
1123,1122,1
1129,564,2
1151,575,2
1153,288,4
1163,166,7
1171,1170,1
1181,236,5
1187,1186,1
1193,298,4
1201,300,4
1213,1212,1
1217,152,8
1223,611,2
1229,1228,1
1231,615,2
1237,1236,1
1249,156,8
1259,1258,1
1277,1276,1
1279,639,2
1283,1282,1
1289,161,8
1291,1290,1
1297,648,2
1301,1300,1
1303,651,2
1307,1306,1
1319,659,2
1321,60,22
1327,221,6
1361,680,2
1367,683,2
1373,1372,1
1381,1380,1
1399,233,6
1409,704,2
1423,237,6
1427,1426,1
1429,84,17
1433,179,8
1439,719,2
1447,723,2
1451,1450,1
1453,1452,1
1459,486,3
1471,245,6
1481,370,4
1483,1482,1
1487,743,2
1489,744,2
1493,1492,1
1499,1498,1
1511,755,2
1523,1522,1
1531,1530,1
1543,771,2
1549,1548,1
1553,194,8
1559,779,2
1567,783,2
1571,1570,1
1579,526,3
1583,791,2
1597,532,3
1601,400,4
1607,803,2
1609,201,8
1613,52,31
1619,1618,1
1621,1620,1
1627,542,3
1637,1636,1
1657,92,18
1663,831,2
1667,1666,1
1669,1668,1
1693,1692,1
1697,848,2
1699,566,3
1709,244,7
1721,215,8
1723,574,3
1733,1732,1
1741,1740,1
1747,1746,1
1753,146,12
1759,879,2
1777,74,24
1783,891,2
1787,1786,1
1789,596,3
1801,25,72
1811,362,5
1823,911,2
1831,305,6
1847,923,2
1861,1860,1
1867,1866,1
1871,935,2
1873,936,2
1877,1876,1
1879,939,2
1889,472,4
1901,1900,1
1907,1906,1
1913,239,8
1931,1930,1
1933,644,3
1949,1948,1
1951,975,2
1973,1972,1
1979,1978,1
1987,1986,1
1993,996,2
1997,1996,1
1999,333,6
2003,286,7
2011,402,5
2017,336,6
2027,2026,1
2029,2028,1
2039,1019,2
2053,2052,1
2063,1031,2
2069,2068,1
2081,1040,2
2083,2082,1
2087,1043,2
2089,29,72
2099,2098,1
2111,1055,2
2113,44,48
2129,532,4
2131,2130,1
2137,1068,2
2141,2140,1
2143,51,42
2153,1076,2
2161,1080,2
2179,726,3
2203,734,3
2207,1103,2
2213,2212,1
2221,2220,1
2237,2236,1
2239,1119,2
2243,2242,1
2251,750,3
2267,2266,1
2269,2268,1
2273,568,4
2281,190,12
2287,381,6
2293,2292,1
2297,1148,2
2309,2308,1
2311,1155,2
2333,2332,1
2339,2338,1
2341,780,3
2347,782,3
2351,47,50
2357,2356,1
2371,2370,1
2377,1188,2
2381,476,5
2383,397,6
2389,2388,1
2393,598,4
2399,1199,2
2411,482,5
2417,1208,2
2423,1211,2
2437,2436,1
2441,305,8
2447,1223,2
2459,2458,1
2467,2466,1
2473,618,4
2477,2476,1
2503,1251,2
2521,1260,2
2531,2530,1
2539,2538,1
2543,1271,2
2549,2548,1
2551,1275,2
2557,2556,1
2579,2578,1
2591,1295,2
2593,81,32
2609,1304,2
2617,1308,2
2621,2620,1
2633,1316,2
2647,1323,2
2657,166,16
2659,2658,1
2663,1331,2
2671,445,6
2677,2676,1
2683,2682,1
2687,79,34
2689,224,12
2693,2692,1
2699,2698,1
2707,2706,1
2711,1355,2
2713,1356,2
2719,1359,2
2729,1364,2
2731,26,105
2741,2740,1
2749,916,3
2753,1376,2
2767,461,6
2777,1388,2
2789,2788,1
2791,465,6
2797,2796,1
2801,1400,2
2803,2802,1
2819,2818,1
2833,118,24
2837,2836,1
2843,2842,1
2851,2850,1
2857,102,28
2861,2860,1
2879,1439,2
2887,1443,2
2897,1448,2
2903,1451,2
2909,2908,1
2917,972,3
2927,1463,2
2939,2938,1
2953,492,6
2957,2956,1
2963,2962,1
2969,371,8
2971,110,27
2999,1499,2
3001,1500,2
3011,3010,1
3019,3018,1
3023,1511,2
3037,3036,1
3041,1520,2
3049,762,4
3061,204,15
3067,3066,1
3079,1539,2
3083,3082,1
3089,772,4
3109,444,7
3119,1559,2
3121,156,20
3137,784,4
3163,1054,3
3167,1583,2
3169,1584,2
3181,1060,3
3187,3186,1
3191,55,58
3203,3202,1
3209,1604,2
3217,804,4
3221,644,5
3229,1076,3
3251,650,5
3253,3252,1
3257,407,8
3259,1086,3
3271,545,6
3299,3298,1
3301,660,5
3307,3306,1
3313,828,4
3319,1659,2
3323,3322,1
3329,1664,2
3331,222,15
3343,557,6
3347,3346,1
3359,1679,2
3361,168,20
3371,3370,1
3373,1124,3
3389,484,7
3391,113,30
3407,1703,2
3413,3412,1
3433,1716,2
3449,431,8
3457,576,6
3461,3460,1
3463,577,6
3467,3466,1
3469,3468,1
3491,3490,1
3499,3498,1
3511,1755,2
3517,3516,1
3527,1763,2
3529,882,4
3533,3532,1
3539,3538,1
3541,236,15
3547,3546,1
3557,3556,1
3559,1779,2
3571,3570,1
3581,3580,1
3583,1791,2
3593,1796,2
3607,601,6
3613,3612,1
3617,1808,2
3623,1811,2
3631,605,6
3637,3636,1
3643,3642,1
3659,3658,1
3671,1835,2
3673,918,4
3677,3676,1
3691,3690,1
3697,1848,2
3701,3700,1
3709,3708,1
3719,1859,2
3727,1863,2
3733,3732,1
3739,534,7
3761,188,20
3767,1883,2
3769,1884,2
3779,3778,1
3793,1896,2
3797,3796,1
3803,3802,1
3821,764,5
3823,637,6
3833,958,4
3847,1923,2
3851,3850,1
3853,3852,1
3863,1931,2
3877,3876,1
3881,388,10
3889,648,6
3907,3906,1
3911,1955,2
3917,3916,1
3919,1959,2
3923,3922,1
3929,1964,2
3931,3930,1
3943,219,18
3947,3946,1
3967,1983,2
3989,3988,1
4001,1000,4
4003,4002,1
4007,2003,2
4013,4012,1
4019,4018,1
4021,4020,1
4027,1342,3
4049,506,8
4051,50,81
4057,169,24
4073,2036,2
4079,2039,2
4091,4090,1
4093,4092,1
Also the factorization of Phi(n,2)

Last fiddled with by sweety439 on 2022-02-05 at 10:14

 2022-02-05, 13:08 #10 kriesel     "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 2·72·67 Posts Hmm, for g = k n + 1, if kn is odd, for kn > 1, g is not a prime factor of anything, since it has a factor 2.
2022-02-05, 13:54   #11
Dr Sardonicus

Feb 2017
Nowhere

22×31×47 Posts

Quote:
 Originally Posted by sweety439 Well, see https://oeis.org/A001917, for any n, all prime factors of Phi(n,2)/gcd(Phi(n,2),n) (where Phi is the cyclotomic polynomial, if n is prime, then this number is the Mersenne number 2^n-1, and if n is power of 2, then this number is the Fermat number 2^(n/2)+1) (they are the primes p such that znorder(Mod(2,p)) = n) are of the form k*n+1
Which has nothing to do with seeking odd prime factors q of 2p - 1 where p is an odd prime. At the risk of arrest for criminal belaboring of the obvious...

We want znorder(2,q) = p. Ergo, ipso facto, ex post facto, ex officio, etcetera etcetera etcetera,

(q-1)/znorder(2,q) = (q-1)/p is even.

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