含義
重一數，就是全是重複數字1的數，比如1111111111111111111。本文為了表示簡單，我們不妨把前面這個數寫作R19，它表示一個由19個1組成的重一數。則，R2=11，R3=111，……以此類推。
有數論基礎的人不難發現，素數個1組成的重一數就是梅桑數底數為10的情況。
重一數看似簡單，但是通過研究重一數的素因子，你可以發現其中隱含著很多有趣的東西。這裡首先引入一個定義——強素因子。
若Rn有素因子p，且對於任何r<n，都不存在Rr有素因子p，我們稱p為Rn的 強素因子。通俗的說，我們從R2、R3、R4開始寫每個重一數的素因子，強素因子就是那些新出現的素因子。可見，若一個素數是Rm的強素因子，他就不可能再是Rn（n≠m）的強素因子了。不同的同一數，它們強素因子所組成的集合之間沒有交集。
兩個定理
定理1
p為素數的充分必要條件是Rp的所有素因子皆為強素因子。
定理2
一個素數p的倒數的循環節位數為m（素數的倒數皆是純循環小數）的充分必要條件是，p為Rm的強素因子。
定理1的例子比如R7=239x4649，而239，4649都是R7的強素因子，換句話說，R2、R3、R4、R5、R6中皆沒有239和4649這兩個素因子。
定理2的例子即1/239=0.<0041841>，這個式子指0041841為1/239的循環節，可見它為7位。
同樣的，1/4649=0.<0002151>，亦為7位循環節。除了239和4649這兩個素數外，再也沒有素數的倒數是7位循環節了。
定理2說明，一個素數的循環節位數表明了它是哪個重一數的強素因子。我們利用電子計算機技術，可以通過計算一個素數的循環節位數，來尋找大重一數（一般是100以上）的強素因子，繼而對該重一數分解素因子。
定理3
Rn的強素因子是kn+1的形式。（k為自然數）
例子比如：239=34x7+1，4649=664x7+1。由定理3可見，Rn的強素因子一定大於n+1，換句話說，一個素數p，它倒數的循環節位數最大為p-1。
對於定理2、3，應該補充一句：對於2、3、5這三個素數例外，很明顯重一數是不可能把2、5作為素因子的，而對於3來說，1/3=0.<3>，只有1位循環節，但是它卻是R3的強素因子，另外3不是3k+1的形式。定理2、3對於其它素數都是滿足的，這一點也很容易證明。
我們可以看到R2是一個素重一數。容易證明，當n為合數時，Rn一定為合數。那么，有沒有更大一些的素重一數呢？答案是有的，本文開頭所示的R19即為第二個素重一數，下一個素重一數是R23，再下一個便是R317了，之後是R1031，R49081，R86453，R109297，R270343。可見，素重一數的分布是沒有規律的。關於素重一數的研究，可以與梅桑素數的研究相似。
形如(10^n-1)/9的整數，如1,11,111,111,...
一般記為Rn.
為素數的前幾個重一數是2, 19, 23, 317, 1031,
49081, 86453, 109297.
前面幾個的分解式
1 no prime factors
2 11
3 3 37
4 (2) 101
5 41 271
6 (2,3) 7 13
7 239 4649
8 (2,4) 73 137
9 (3) 3 333667
10 (2,5) 9091
11 21649 513239
12 (2,3,4,6) 9901
13 53 79 265371653
14 (2,7) 909091
15 (3,5) 31 2906161
16 (2,4,8) 17 5882353
17 2071723 5363222357
18 (2,3,6,9) 19 52579
19 1111111111111111111
20 (2,4,5,10) L M
L 3541
M 27961
21 (3,7) 43 1933 10838689
22 (2,11) 11 23 4093 8779
23 11111111111111111111111
24 (2,3,4,6,8,12) 99990001
25 (5) 21401 25601 182521213001
26 (2,13) 859 1058313049
27 (3,9) 3 757 440334654777631
28 (2,4,7,14) 29 281 121499449
29 3191 16763 43037 62003 77843839397
30 (2,3,5,6,10,15) 211 241 2161
31 2791 6943319 57336415063790604359
32 (2,4,8,16) 353 449 641 1409 69857
33 (3,11) 67 1344628210313298373
34 (2,17) 103 4013 21993833369
35 (5,7) 71 123551 102598800232111471
36 (2,3,4,6,9,12,18) 999999000001
37 2028119 247629013 2212394296770203368013
38 (2,19) 909090909090909091
39 (3,13) 900900900900990990990991
40 (2,4,5,8,10,20) 1676321 5964848081
41 83 1231 538987 201763709900322803748657942361
42 (2,3,6,7,14,21) 7 127 2689 459691
43 173 1527791 1963506722254397 2140992015395526641
44 (2,4,11,22) 89 1052788969 1056689261
45 (3,5,9,15) 238681 4185502830133110721
46 (2,23) 47 139 2531 549797184491917
47 35121409 316362908763458525001406154038726382279
48 (2,3,4,6,8,12,16,24) 9999999900000001
49 (7) 505885997 1976730144598190963568023014679333
50 (2,5,10,25) 251 5051 78875943472201
51 (3,17) 613 210631 52986961 13168164561429877
52 (2,4,13,26) 521 1900381976777332243781
53 107 1659431 1325815267337711173
47198858799491425660200071
54 (2,3,6,9,18,27) 70541929 14175966169
55 (5,11) 1321 62921 83251631 1300635692678058358830121
56 (2,4,7,8,14,28) 7841 127522001020150503761
57 (3,19) 21319 10749631 3931123022305129377976519
58 (2,29) 59 154083204930662557781201849
59 2559647034361
4340876285657460212144534289928559826755746751
60 (2,3,4,5,6,10,12,15,30) L M
L (20L) 61 4188901
M (20M) 39526741
61 733 4637 329401 974293 1360682471 106007173861643
7061709990156159479
62 (2,31) 909090909090909090909090909091
63 (3,7,9,21) 10837 23311 45613 45121231 1921436048294281
64 (2,4,8,16,32) 19841 976193 6187457 834427406578561
65 (5,13) 162503518711 5538396997364024056286510640780600481
66 (2,3,6,11,22,33) 599144041 183411838171
67 493121 79863595778924342083
28213380943176667001263153660999177245677
68 (2,4,17,34) 28559389 1491383821 2324557465671829
69 (3,23) 277 203864078068831 1595352086329224644348978893
70 (2,5,7,10,14,35) 4147571 265212793249617641
71 241573142393627673576957439049
45994811347886846310221728895223034301839
72 (2,3,4,6,8,9,12,18,24,36) 3169 98641 3199044596370769
73 12171337159 1855193842151350117
49207341634646326934001739482502131487446637
74 (2,37) 7253 422650073734453 296557347313446299
75 (3,5,15,25) 151 4201 15763985553739191709164170940063151
76 (2,4,19,38) 722817036322379041 1369778187490592461
77 (7,11) 5237 42043 29920507
136614668576002329371496447555915740910181043
78 (2,3,6,13,26,39) 13 157 6397 216451 388847808493
79 317 6163 10271 307627 49172195536083790769
3660574762725521461527140564875080461079917
80 (2,4,5,8,10,16,20,40) 5070721 19721061166646717498359681
81 (3,9,27) 3 163 9397 2462401 676421558270641
130654897808007778425046117
82 (2,41) 2670502781396266997 3404193829806058997303
83 3367147378267 9512538508624154373682136329
346895716385857804544741137394505425384477
84 (2,3,4,6,7,12,14,21,28,42) 226549 4458192223320340849
85 (5,17) 262533041 8119594779271
4222100119405530170179331190291488789678081
86 (2,43) 57009401 2182600451 7306116556571817748755241
87 (3,29) 4003 72559
310170251658029759045157793237339498342763245483
88 (2,4,8,11,22,44) 617
16205834846012967584927082656402106953
89 497867 103733951 104984505733 5078554966026315671444089
403513310222809053284932818475878953159
90 (2,3,5,6,9,10,15,18,30,45) 29611 3762091 8985695684401
91 (7,13) 547 14197 17837 4262077 43442141653
316877365766624209 110742186470530054291318013
92 (2,4,23,46) 1289 18371524594609
4181003300071669867932658901
93 (3,31)
900900900900900900900900900900990990990990990990990990990991
94 (2,47) 6299 4855067598095567 297262705009139006771611927
95 (5,19) 191 59281 63841 1289981231950849543985493631
965194617121640791456070347951751
96 (2,3,4,6,8,12,16,24,32,48) 97 206209 66554101249
75118313082913
97 12004721 846035731396919233767211537899097169
109399846855370537540339266842070119107662296580348039
98 (2,7,14,49) 197 5076141624365532994918781726395939035533
99 (3,9,11,33) 199 397 34849
362853724342990469324766235474268869786311886053883
100 (2,4,5,10,20,25,50) L M
L 7019801 14103673319201
M 60101 1680588011350901
101 4531530181816613234555190841
129063282232848961951985354966759
18998088572819375252842078421374368604969
102 (2,3,6,17,34,51) 291078844423 377526955309799110357
103 1031 7034077
153211620887015423991278431667808361439217294295901387715486 $
473457925534859044796980526236853
104 (2,4,8,13,26,52) 1580801
632527440202150745090622412245443923049201
105 (3,5,7,15,21,35) 30703738801 625437743071
57802050308786191965409441
106 (2,53) 9090909090909090909090909090909090909090909090909091
107 643 999809 9885089 215257037 2386760191
511399538427507881 646826950155548399
10288079467222538791302311556310051849
108 (2,3,4,6,9,12,18,27,36,54) 109 153469
59779577156334533866654838281
109 1192679 712767480971213008079 5295275348767234696493
246829743984355435962408390910378218537282105150086881669547
110 (2,5,10,11,22,55) 331 5171 20163494891
318727841165674579776721
111 (3,37) 37 30557051518647307 8845981170865629119271997
90077814396055017938257237117
112 (2,4,7,8,14,16,28,56) 113 73765755896403138401
119968369144846370226083377
113 227 908191467191
538957123122177190652671034266853972984987051734492265550033 $
46881878523705781079015749721646701723
114 (2,3,6,19,38,57) 1458973 753201806271328462547977919407
115 (5,23) 31511 19707665921 20414137203567631
5799951513941382144830754391 122403569491783662720773144041
116 (2,4,29,58) 349 38861 618049
11811806375201836408679635736258669583187541
117 (3,9,13,39) 240396841140769 537947698126879
3352825314499987 2304017384484085131816292573
118 (2,59) 1889 1090805842068098677837
4411922770996074109644535362851087
119 (7,17) 923441 3924966376871
768736559421401249042753476963
323012942148562751650814544437350454640448842187
120 (2,3,4,5,6,8,10,12,15,20,24,30,40,60)
100009999999899989999000000010001
