mersenneforum.org (New ?) Wagstaff/Mersenne related property
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 2019-11-22, 15:32 #1 T.Rex     Feb 2004 France 22·229 Posts (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
 2019-11-23, 00:51 #2 GP2     Sep 2003 2·5·7·37 Posts 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
 2019-11-23, 21:45 #3 sweety439   Nov 2016 2,819 Posts 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 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 250 229668251.5519485418336288303251 251 503.54217.178230287214063289511.61676882198695257501367.P26 252L 118750098349 252M 40388473189 253 23*.4103188409.199957736328435366769577.P32 254 P38 255 106591.949111.P28 256 59649589127497217.5704689200685129054721 257 535006138814359.1155685395246619182673033.P39 258 1033.1591582393.15686603697451 259 2499285769.P56 260L 108140989558681 260M 521.51481.34110701 261 P51 262 1049.4744297.P30 263 23671.13572264529177.120226360536848498024035943.P36 264 7393.1761345169.98618273953 265 29324808311.197748738449921.P38 266 4523.P30 267 78903841.28753302853087.P32 268L 269.42875177.2559066073 268M 15152453.9739278030221 269 13822297.P74 270 811.15121.385838642647891 271 15242475217.P72 272 383521.2368179743873.373200722470799764577 273 108749551.4093204977277417.86977595801949844993 274 1097.15619.32127963626435681.105498212027592977 275 382027665134363932751.P40 276L 70334392823809 276M 5415624023749 277 1121297.31133636305610209482201109050392404721.P40 278 4506937.P35 279 16183.34039.1437967.833732508401263.2034439836951867299888617 280 84179842077657862011867889681 281 80929.P80 282 1681003.35273039401.111349165273 283 9623.68492481833.P71 284L 4999465853.472287102421 284M 569.148587949.5585522857 285 1491477035689218775711.25349242986637720573561 286 2003.6156182033.10425285443.15500487753323 287 17137716527.P62 288 1153.6337.38941695937.278452876033 289 12761663.179058312604392742511009.P52 290 7553921.999802854724715300883845411 291 272959.2065304407.5434876633.P34 292L 9444732965601851473921 292M 293.9929.649301712182209 293 40122362455616221971122353.P63 294 748819.26032885845392093851 295 4721.132751.5794391.128818831.3812358161.452824604065751.P22 296 P44 297 8950393.P48 298 1193.650833.38369587.7984559573504259856359124657 299 599.9341359.14718679249.13444476836590589479.51441563151591093599.260242449712509916159 300L 63901.13334701 300M 1201.1182468601 301 490631.365505823711978039310711.P47 302 18717738334417.P32 303 607.P58 304 27361.69394460463940481.11699557817717358904481 305 1831.2441.4271.270841.484074637694471.P42 306 123931.26159806891.27439122228481 307 14608903.85798519.23487583303.78952752017.P57 308L 869467061.3019242689 308M 8317.76096559910757 309 1953272766780718501831.P40 310 11161.5947603221397891.29126056043168521 311 5344847.2647649373910205158468946067671.P57 312 84159375948762099254554456081 313 10960009.14787970697180273.3857194764289141165278097.P47 314 15073.2350291.17751783757817897.96833299198971305921 315 870031.983431.P32 316L 604462909806215075725313 316M 317.381364611866507317969 317 9511.587492521482839879.4868122671322098041565641.P49 318 6043.4475130366518102084427698737 319 18503.64439.84819793631.P64 320 3602561.P32 321 17866285599391.P51 322 P40 323 647.7753.39044358788825633753.P61 324L 3618757.4977454861 324M 106979941.168410989 325 7151.51879585551.P58 326 11281292593.1023398150341859.337570547050390415041769 327 20597276734348736647.33157029794959983067039.P23 328 13121.8562191377.P35 329 12503.200033.9106063.270447871.P58 330 415365721.2252127523412251 331 16937389168607.865118802936559.P72 332L 13063537.148067197374074653 332M 997.46202197673.209957719973 333 1999.10657.169831.1238761.36085879.199381087.698962539799.4096460559560875111 334 P50 335 464311.1532217641.P65 336 2017.25629623713.1538595959564161 337 18199.2806537.95763203297.726584894969.P68 338 4929910764223610387.18526238646011086732742614043 339 10113049.320021624768405574452943847.P34 340L 1021.4421.550801.23650061 340M 7226904352843746841 341 5560125493425335999.126901141805369975317583.P49 342 19*.P32 343 6073159.1428389887.62228099977.P62 344 3855260977.64082150767423457.1425343275103126327372769 345 P54 346 347.4153.35374479827.47635010587.1643464247728189221623609 347 14143189112952632419639.P83 348L 22170214192500421 348M 349.29581.27920807689 349 1779973928671.34720396273212657799920861294559.P62 350 1051.110251.347833278451.34010032331525251 351 446473.29121769.571890896913727.93715008807883087.P21 352 5304641.P42 353 931921.2927455476800301964116805545194017.P67 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2019-11-23, 21:46   #4
sweety439

Nov 2016

2,819 Posts

Quote:
 Originally Posted by T.Rex 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

 2019-11-23, 22:25 #5 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 23×11×23 Posts 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
2019-11-23, 22:39   #6
sweety439

Nov 2016

2,819 Posts

Quote:
 Originally Posted by a1call 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

 2019-11-23, 22:46 #7 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 111111010002 Posts The Mersenne composite must have a prime exponent. Is 170 a prime number?

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