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-   -   (New ?) Wagstaff/Mersenne related property (https://www.mersenneforum.org/showthread.php?t=24964)

T.Rex 2019-11-22 15:32

(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 [TEX]q[/TEX] prime [TEX]>3[/TEX]
[TEX]q=2p+1[/TEX] and thus [TEX]p=\frac{q-1}{2}[/TEX].

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

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

Then:
[TEX]2N_pM_p+3 = 2^q-2+3 = 2^q+1 = 3W_q[/TEX]

Thus the [B]property [/B]: [B][TEX]W_q = \frac{2}{3}N_pM_p+1[/TEX][/B] . CQFD.

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

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


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

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

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

GP2 2019-11-23 00:51

Renaud and Henri Lifchitz [URL="http://www.primenumbers.net/Documents/TestNP.pdf"]mentioned this relation in a 2000 paper[/URL], see section 4. Note: they use N[SUB]p[/SUB] to denote what we call W[SUB]p[/SUB].

As they point out, this relation means that if W[SUB]2n+1[/SUB] is PRP, then if either W[SUB]n[/SUB] or M[SUB]n[/SUB] are fully factored, then W[SUB]2n+1[/SUB] 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 M[SUB]47684[/SUB] or W[SUB]47684[/SUB] then we could prove that W[SUB]95369[/SUB] is not just PRP but prime. Spoiler alert: they are nowhere near fully-factored.

Or we could look at all Mersenne primes M[SUB]p[/SUB] for which 2p+1 is also prime ([URL="https://oeis.org/A065406"]OEIS A065406[/URL]) and check to see if any unfactored W[SUB]2p+1[/SUB] test PRP. Spoiler alert: they don't. M[SUB]43,112,609[/SUB] is a Mersenne prime but W[SUB]86,225,219[/SUB] is composite.

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

sweety439 2019-11-23 21:45

p divides M[SUB]n[/SUB] if and only if ord[SUB]p[/SUB](2) divides n, and p divides W[SUB]n[/SUB] (for odd n) if and only if ord[SUB]p[/SUB](2) is even and divides 2n

This is the list for ord[SUB]p[/SUB](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
[/CODE]

This is the factorization of Phi[SUB]n[/SUB](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
243 487.16753783618801.192971705688577.3712990163251158343
244L 733.1709.368140581013
244M 3456749.667055378149
245 1471.P48
246 739.165313.13194317913029593
247 15809.6459570124697.402004106269663.P34
248 290657.3770202641.1141629180401976895873
249 1621324657.P40
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[/CODE]

sweety439 2019-11-23 21:46

[QUOTE=T.Rex;531257]Hi,

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

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

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

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

Then:
[TEX]2N_pM_p+3 = 2^q-2+3 = 2^q+1 = 3W_q[/TEX]

Thus the [B]property [/B]: [B][TEX]W_q = \frac{2}{3}N_pM_p+1[/TEX][/B] . CQFD.

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

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


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

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

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

if p is prime, q=2*p+1 is also prime, then:

q divides M[SUB]p[/SUB] if and only if p = 3 mod 4

q divides W[SUB]p[/SUB] if and only if p = 1 mod 4

a1call 2019-11-23 22:25

The Mersenne part has been known since 1775 as proven in this link first shown to me by sm:
[url]https://primes.utm.edu/notes/proofs/MerDiv2.html[/url]
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

sweety439 2019-11-23 22:39

[QUOTE=a1call;531350]The Mersenne part has been known since 1775 as proven in this link first shown to me by sm:
[url]https://primes.utm.edu/notes/proofs/MerDiv2.html[/url]
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[/QUOTE]

341 divides M170 and 341=2*170+1, but 341 is not prime :cmd::cmd::cmd:

a1call 2019-11-23 22:46

The Mersenne composite must have a prime exponent.
Is 170 a prime number?


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