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Old 2019-11-22, 15:32   #1
T.Rex
 
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Feb 2004
France

2·457 Posts
Default (New ?) Wagstaff/Mersenne related property

Hi,

I have no idea if this property is new. If new, I even am not sure it may be useful.
Anyway.

Let q prime >3
q=2p+1 and thus p=\frac{q-1}{2}.

Let:
N_p=2^p+1 .
M_p=2^p-1 Mersenne.
N_p M_p = 2^{2p}-1=2^{q-1}-1

Let:
W_q=\frac{2^q+1}{3} Wagstaff.

Then:
2N_pM_p+3 = 2^q-2+3 = 2^q+1 = 3W_q

Thus the property : W_q = \frac{2}{3}N_pM_p+1 . CQFD.

\alpha \mid W_q \Rightarrow \alpha = 1+2q\alpha'
thus : W_q = 1+2q\beta and 2q\beta = W_q-1 = \frac{2}{3}N_pM_p
thus : q \, \mid \, \frac{N_pM_p}{3} and thus either q \mid N_p or q \mid M_p .

Examples :
q=11 , \, p=5 , \, q \mid N_p
q=17 , \, p=8 , \, q \mid M_p
q=47 , \, p=23 , \, q \mid M_p
q=59 , \, p=29 , \, q \mid N_p
q=257 , \, p=2^7 , \, q \mid M_p
q=65537 , \, p=2^{15} , \, q \mid ?_p


Probably one should only consider cases where p is a prime or a power of 2.

If p = 2^n, then 3 divides M_p since only numbers k.2^{n+1}+1 can divide a Fermat number. Can k be 1 ?

If p is a prime, thus 3 cannot divide M_p since only numbers 1+\alpha p can divide a Mersenne number, and thus 3 divides N_p.
So, when p is a prime, when does it divide N_p and not M_p and vice-versa ??

Last fiddled with by T.Rex on 2019-11-22 at 16:24
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Old 2019-11-23, 00:51   #2
GP2
 
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Sep 2003

22·3·5·43 Posts
Default

Renaud and Henri Lifchitz mentioned this relation in a 2000 paper, see section 4. Note: they use Np to denote what we call Wp.

As they point out, this relation means that if W2n+1 is PRP, then if either Wn or Mn are fully factored, then W2n+1 can be proven prime by the N−1 method. (Note n does not need to be prime, only 2n+1).

For instance, if we could fully factor M47684 or W47684 then we could prove that W95369 is not just PRP but prime. Spoiler alert: they are nowhere near fully-factored.

Or we could look at all Mersenne primes Mp for which 2p+1 is also prime (OEIS A065406) and check to see if any unfactored W2p+1 test PRP. Spoiler alert: they don't. M43,112,609 is a Mersenne prime but W86,225,219 is composite.

In short, this relation doesn't have much practical use.

Last fiddled with by GP2 on 2019-11-23 at 01:01
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Old 2019-11-23, 21:45   #3
sweety439
 
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Nov 2016

1000101111112 Posts
Default

p divides Mn if and only if ordp(2) divides n, and p divides Wn (for odd n) if and only if ordp(2) is even and divides 2n

This is the list for ordp(2) for small odd primes:

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

Code:
1      
2      3
3      7
4L      
4M      5
5      31
6      3*
7      127
8      17
9      73
10      11
11      23.89
12L      
12M      13
13      8191
14      43
15      151
16      257
17      131071
18      3*.19
19      524287
20L      5*
20M      41
21      7*.337
22      683
23      47.178481
24      241
25      601.1801
26      2731
27      262657
28L      113
28M      29
29      233.1103.2089
30      331
31      2147483647
32      65537
33      599479
34      43691
35      71.122921
36L      37
36M      109
37      223.616318177
38      174763
39      79.121369
40      61681
41      13367.164511353
42      5419
43      431.9719.2099863
44L      397
44M      2113
45      631.23311
46      2796203
47      2351.4513.13264529
48      97.673
49      4432676798593
50      251.4051
51      103.2143.11119
52L      1613
52M      53.157
53      6361.69431.20394401
54      3*.87211
55      881.3191.201961
56      15790321
57      32377.1212847
58      59.3033169
59      179951.3203431780337
60L      61
60M      1321
61      2305843009213693951
62      715827883
63      92737.649657
64      641.6700417
65      145295143558111
66      67.20857
67      193707721.761838257287
68L      137.953
68M      26317
69      10052678938039
70      281.86171
71      228479.48544121.212885833
72      433.38737
73      439.2298041.9361973132609
74      1777.25781083
75      100801.10567201
76L      229.457
76M      525313
77      581283643249112959
78      22366891
79      2687.202029703.1113491139767
80      4278255361
81      2593.71119.97685839
82      83.8831418697
83      167.57912614113275649087721
84L      14449
84M      1429
85      9520972806333758431
86      2932031007403
87      4177.9857737155463
88      353.2931542417
89      618970019642690137449562111
90      18837001
91      911.112901153.23140471537
92L      277.30269
92M      1013.1657
93      658812288653553079
94      283.165768537521
95      191.420778751.30327152671
96      193.22253377
97      11447.13842607235828485645766393
98      4363953127297
99      199.153649.33057806959
100L      101.8101
100M      5*.268501
101      7432339208719.341117531003194129
102      307.2857.6529
103      2550183799.3976656429941438590393
104      858001.308761441
105      29191.106681.152041
106      107.28059810762433
107      P33
108L      246241
108M      279073
109      745988807.870035986098720987332873
110      11*.2971.48912491
111      321679.26295457.319020217
112      5153.54410972897
113      3391.23279.65993.1868569.1066818132868207
114      571.160465489
115      14951.4036961.2646507710984041
116L      107367629
116M      536903681
117      937.6553.86113.7830118297
118      2833.37171.1824726041
119      239.20231.62983048367.131105292137
120      4562284561
121      727.P31
122      768614336404564651
123      3887047.177722253954175633
124L      5581.384773
124M      8681.49477
125      269089806001.4710883168879506001
126      77158673929
127      P39
128      274177.67280421310721
129      11053036065049294753459639
130      131.409891.7623851
131      263.P38
132L      312709
132M      4327489
133      P33
134      7327657.6713103182899
135      271.348031.49971617830801
136      17*.354689.2879347902817
137      32032215596496435569.5439042183600204290159
138      139.168749965921
139      5625767248687.P30
140L      47392381
140M      7416361
141      4375578271.646675035253258729
142      56409643.13952598148481
143      724153.158822951431.5782172113400990737
144      577.487824887233
145      P34
146      1753.1795918038741070627
147      7*.2741672362528725535068727
148L      149.184481113
148M      593.231769777
149      86656268566282183151.8235109336690846723986161
150      1133836730401
151      18121.55871.165799.2332951.7289088383388253664437433
152      1217.148961.24517014940753
153      919.75582488424179347083438319
154      617.78233.35532364099
155      31*.311.11471.73471.4649919401.18158209813151
156L      13*.313.1249
156M      3121.21841
157      852133201.60726444167.1654058017289.2134387368610417
158      201487636602438195784363
159      6679.13960201.540701761.229890275929
160      414721.44479210368001
161      1289.3188767.45076044553.14808607715315782481
162      3*.163.135433.272010961
163      150287.704161.110211473.27669118297.36230454570129675721
164L      181549.12112549
164M      10169.43249589
165      2048568835297380486760231
166      499.1163.2657.155377.13455809771
167      2349023.P44
168      3361.88959882481
169      4057.6740339310641.P31
170      26831423036065352611
171      93507247.3042645634792541312037847
172L      1759217765581
172M      173.101653.500177
173      730753.1505447.70084436712553223.155285743288572277679887
174      96076791871613611
175      39551.60816001.535347624791488552837151
176      229153.119782433.43872038849
177      184081.27989941729.9213624084535989031
178      179.62020897.18584774046020617
179      359.1433.P49
180L      181.54001
180M      29247661
181      43441.1164193.7648337.P37
182      224771.1210483.25829691707
183      367.55633.37201708625305146303973352041
184      291280009243618888211558641
185      1587855697992791.7248808599285760001152755641
186      529510939.2903110321
187      707983.P43
188L      140737471578113
188M      3761.7484047069
189      1560007.207617485544258392970753527
190      2281.3011347479614249131
191      383.7068569257.39940132241.332584516519201.87274497124602996457
192      18446744069414584321
193      13821503.61654440233248340616559.14732265321145317331353282383
194      971.1553.31817.1100876018364883721
195      P30
196L      4981857697937
196M      197.19707683773
197      7487.P56
198      5347.242099935645987
199      164504919713.P49
200      401.340801.2787601.3173389601
201      1609.22111.P32
202      P30
203      136417.121793911.P38
204L      409.3061.13669
204M      1326700741
205      2940521.70171342151.P31
206      415141630193.8142767081771726171
207      79903.634569679.2232578641663.42166482463639
208      78919881726271091143763623681
209      94803416684681.1512348937147247.5346950541323960232319657
210      211.664441.1564921
211      15193.60272956433838849161.P40
212L      1801439824104653
212M      15358129.586477649
213      66457.2849881972114740679.4205268574191396793
214      643.84115747449047881488635567801
215      1721.731516431.514851898711.297927289744047764444862191
216      33975937.138991501037953
217      5209.62497.6268703933840364033151.378428804431424484082633
218      104124649.2077756847362348863128179
219      3943.671165898617413417.4815314615204347717321
220L      415878438361
220M      3630105520141
221      1327.P55
222      3331.17539.107775231312019
223      18287.196687.1466449.2916841.1469495262398780123809.P24
224      449.2689.183076097.358429848460993
225      115201.617401.1348206751.13861369826299351
226      227.48817.636190001.491003369344660409
227      26986333437777017.P52
228L      131101.160969
228M      275415303169
229      1504073.20492753.59833457464970183.P39
230      691.1884103651.345767385170491
231      463.P34
232      59393.82280195167144119832390568177
233      1399.135607.622577.P57
234      5302306226370307681801
235      2391314881.72296287361.P35
236L      1181.3541.157649.174877
236M      5521693.104399276341
237      1423.49297.23728823512345609279.31357373417090093431
238      823679683.143162553165560959297
239      479.1913.5737.176383.134000609.P49
240      394783681.46908728641
241      22000409.P66
242      117371.11054184582797800455736061107
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Old 2019-11-23, 21:46   #4
sweety439
 
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Quote:
Originally Posted by T.Rex View Post
Hi,

I have no idea if this property is new. If new, I even am not sure it may be useful.
Anyway.

Let q prime >3
q=2p+1 and thus p=\frac{q-1}{2}.

Let:
N_p=2^p+1 .
M_p=2^p-1 Mersenne.
N_p M_p = 2^{2p}-1=2^{q-1}-1

Let:
W_q=\frac{2^q+1}{3} Wagstaff.

Then:
2N_pM_p+3 = 2^q-2+3 = 2^q+1 = 3W_q

Thus the property : W_q = \frac{2}{3}N_pM_p+1 . CQFD.

\alpha \mid W_q \Rightarrow \alpha = 1+2q\alpha'
thus : W_q = 1+2q\beta and 2q\beta = W_q-1 = \frac{2}{3}N_pM_p
thus : q \, \mid \, \frac{N_pM_p}{3} and thus either q \mid N_p or q \mid M_p .

Examples :
q=11 , \, p=5 , \, q \mid N_p
q=17 , \, p=8 , \, q \mid M_p
q=47 , \, p=23 , \, q \mid M_p
q=59 , \, p=29 , \, q \mid N_p
q=257 , \, p=2^7 , \, q \mid M_p
q=65537 , \, p=2^{15} , \, q \mid ?_p


Probably one should only consider cases where p is a prime or a power of 2.

If p = 2^n, then 3 divides M_p since only numbers k.2^{n+1}+1 can divide a Fermat number. Can k be 1 ?

If p is a prime, thus 3 cannot divide M_p since only numbers 1+\alpha p can divide a Mersenne number, and thus 3 divides N_p.
So, when p is a prime, when does it divide N_p and not M_p and vice-versa ??
if p is prime, q=2*p+1 is also prime, then:

q divides Mp if and only if p = 3 mod 4

q divides Wp if and only if p = 1 mod 4
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Old 2019-11-23, 22:25   #5
a1call
 
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The Mersenne part has been known since 1775 as proven in this link first shown to me by sm:
https://primes.utm.edu/notes/proofs/MerDiv2.html
It is of practical use in proving factors of Mersenne numbers with prime exponents prime.
23 is guaranteed to be prime because it divides M11 and 23= 2*11+1

Last fiddled with by a1call on 2019-11-23 at 22:26
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Old 2019-11-23, 22:39   #6
sweety439
 
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Quote:
Originally Posted by a1call View Post
The Mersenne part has been known since 1775 as proven in this link first shown to me by sm:
https://primes.utm.edu/notes/proofs/MerDiv2.html
It is of practical use in proving factors of Mersenne numbers with prime exponents prime.
23 is guaranteed to be prime because it divides M11 and 23= 2*11+1
341 divides M170 and 341=2*170+1, but 341 is not prime
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Old 2019-11-23, 22:46   #7
a1call
 
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The Mersenne composite must have a prime exponent.
Is 170 a prime number?
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