HOME plateWON | World!OfNumbers Repunits and their prime factors Normal Smarandache Concatenated Numbers Reversed Smarandache Concatenated Numbers Circular Primes

Introduction

In the table below you'll find all the prime factors of the first 100 repunits.
The first not yet factored repunit seems to be R353 with a composite of 328 digits.
Makoto Kamada, Factorization of 11...11 (repunits).

Sources

Some repunits prime factor lists on the web

Listing 1 - Repunit100 - date.pdf from Koide Yousuke
Listing 2 - Factors of 10^n – 1, n odd, n ⩽ 999 from Torbjörn Granlund
Listing 3 - Factorization of 10^n – 1, n odd, n < 450 by Sam Wagstaff.
Listing 4 - Cunningham Book, Third Edition, Online version

Some interesting repunits websources

Factorization of Repunit Numbers - Repunit_factors.txt, up to n=8580 ! (archived version) by Koide Yousuke from Japan.
Repunit by Eric W. Weisstein
The Cunningham Project by Sam Wagstaff
OEIS Sequence A000042 - Unary representation of natural numbers.
OEIS Sequence A002275 - Repunits: (10^n – 1)/9.
OEIS Sequence A004023 - Indices of prime "repunits": numbers n such that 11...111 = (10^n – 1)/9 is prime.
OEIS Sequence A067063 - Smallest prime factor of repunit(n) = (10^n – 1)/9.
OEIS Sequence A003020 - Largest factor of the "repunit" number 11...1.

Some repunits archived 'Yahoo.Groups.Primeform' messages about R(86453)

Message 1128 Message 1132 Message 1133 Message 1134 Message 1139 Message 1140 Message 1141 Message 1142 Message 1143 Message 1144

New PRP Repunit R(270343) from 15 Jul 2007 by Maksym Voznyy & Anton Budnyy
New Repunit R(109297) from 3 Apr 2007 by Harvey Dubner
R86453 is a New Probable Prime Repunit from 26 Oct 2000 by Lew Baxter
New prp Repunit, R(49081) from 9 Sep 1999 by Harvey Dubner

Messages (from most recent down to oldest)

[ May 15, 2023 ]
Message forwarded to me by Xinyao (Richard) Chen (email).

The repunit R(86453) is certificated to be prime, see announcement by Andreas Enge at https://www.mersenneforum.org/showpost.php?p=630711&postcount=236
Specifically last entry at https://www.multiprecision.org/cm/ecpp.html

[ April 20, 2022 ]
Message forwarded to me by Xinyao (Richard) Chen (email).

The repunit R(49081) is now proven prime by Paul Underwood [ March 21, 2022 ], see https://www.mersenneforum.org/showpost.php?p=602219&postcount=35

Also, the repunit R(323) was factored, see http://pastebin.com/k9ahZHbp, currently the first not yet factored repunit is R(353), with a composite of 328 digits.

[ May 8, 2021 ]
Serge Batalov and Ryan Propper announce a [ New Probable Prime Repunit, R(8177207) ]

References

Serge Batalov "New repunit (PRP) prime found, 8177207 decimal digits, a PRP record".

Sloane's OEIS sequence A004023 - Indices of prime "repunits": numbers n such that 11...111 = (10^n – 1)/9 is prime.

[ April 20, 2021 ]
Serge Batalov and Ryan Propper announce a [ New Probable Prime Repunit, R(5794777) ]

References

Serge Batalov “New repunit (PRP) prime found, 5794777 decimal digits, a PRP record”.

Sloane's OEIS sequence A004023 - Indices of prime "repunits": numbers n such that 11...111 = (10^n – 1)/9 is prime.

[ July 11, 2007 ]
Maksym Voznyy and Anton Budnyy (email) announce a New Probable Prime Repunit, R(270343)

References

Voznyy, M. “New PRP Repunit R(270343)” NMBRTHRY@LISTSERV.NODAK.EDU mailing list. 15 Jul 2007.
https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;1a44c5be.0707&S=

[ April 3, 2007 ]
Harvey Dubner announces a New Probable Prime Repunit, R(109297)

[ January 9, 2006 ]
Evgenii Meerson (email) from Jerusalem (Israel) recently wrote a short note where
new results about the repunits are presented.
See 'Repunits and Harmony.pdf' in his site at https://web.archive.org/web/20190710052947/http://evgenii-meerson-pr.tripod.com/.
Evgenii hopes that his results may be useful for everyone showing interest in the
repunit properties.

[ May 20, 2001 ] Last update [ June 18, 2022 ]

Hans Havermann (email) provides now a list with prime factors up to repunit R(352).
Hans' website is at Factorization of Repunits, n < 353

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The Factorization Table

R111 trivial case
R2Is Prime!1
R33
37
2
R411
101
2
R541
271
2
R63
7
11
13
37
5
R7239
4.649
2
R811
73
101
137
4
R93
3
37
333.667
4
R1011
41
271
9.091
4
R1121.649
513.239
2
R123
7
11
13
37
101
9.901
7
R1353
79
265.371.653
3
R1411
239
4.649
909.091
4
R153
31
37
41
271
2.906.161
6
R1611
17
73
101
137
5.882.353
6
R172.071.723
5.363.222.357
2
R183
3
7
11
13
19
37
52.579
333.667
9
R19Is Prime!1
R2011
41
101
271
3.541
9.091
27.961
7
R213
37
43
239
1.933
4.649
10.838.689
7
R2211
11
23
4.093
8.779
21.649
513.239
7
R23Is Prime!1
R243
7
11
13
37
73
101
137
9.901
99.990.001
10
R2541
271
21.401
25.601
182.521.213.001
5
R2611
53
79
859
265.371.653
1.058.313.049
6
R273
3
3
37
757
333.667
440.334.654.777.631
7
R2811
29
101
239
281
4.649
909.091
121.499.449
8
R293.191
16.763
43.037
62.003
77.843.839.397
5
R303
7
11
13
31
37
41
211
241
271
2.161
9.091
2.906.161
13
R312.791
6.943.319
57.336.415.063.790.604.359
3
R3211
17
73
101
137
353
449
641
1.409
69.857
5.882.353
11
R333
37
67
21.649
513.239
1.344.628.210.313.298.373
6
R3411
103
4.013
2.071.723
5.363.222.357
21.993.833.369
6
R3541
71
239
271
4.649
123.551
102.598.800.232.111.471
7
R363
3
7
11
13
19
37
101
9.901
52.579
333.667
999.999.000.001
12
R372.028.119
247.629.013
2.212.394.296.770.203.368.013
3
R3811
909.090.909.090.909.091
R19
3
R393
37
53
79
265.371.653
900.900.900.900.990.990.990.991
6
R4011
41
73
101
137
271
3.541
9.091
27.961
1.676.321
5.964.848.081
11
R4183
1.231
538.987
201.763.709.900.322.803.748.657.942.361
4
R423
7
7
11
13
37
43
127
239
1.933
2.689
4.649
459.691
909.091
10.838.689
15
R43173
1.527.791
1.963.506.722.254.397
2.140.992.015.395.526.641
4
R4411
11
23
89
101
4.093
8.779
21.649
513.239
1.052.788.969
1.056.689.261
11
R453
3
31
37
41
271
238.681
333.667
2.906.161
4.185.502.830.133.110.721
10
R4611
47
139
2.531
549.797.184.491.917
R23
6
R4735.121.409
316.362.908.763.458.525.001.406.154.038.726.382.279
2
R483
7
11
13
17
37
73
101
137
9.901
5.882.353
99.990.001
9.999.999.900.000.001
13
R49239
4.649
505.885.997
1.976.730.144.598.190.963.568.023.014.679.333
4
R5011
41
251
271
5.051
9.091
21.401
25.601
182.521.213.001
78.875.943.472.201
10
R513
37
613
210.631
2.071.723
52.986.961
5.363.222.357
13.168.164.561.429.877
8
R5211
53
79
101
521
859
265.371.653
1.058.313.049
1.900.381.976.777.332.243.781
9
R53107
1.659.431
1.325.815.267.337.711.173
47.198.858.799.491.425.660.200.071
4
R543
3
3
7
11
13
19
37
757
52.579
333.667
70.541.929
14.175.966.169
440.334.654.777.631
14
R5541
271
1.321
21.649
62.921
513.239
83.251.631
1.300.635.692.678.058.358.830.121
8
R5611
29
73
101
137
239
281
4.649
7.841
909.091
121.499.449
127.522.001.020.150.503.761
12
R573
37
21.319
10.749.631
R19
3.931.123.022.305.129.377.976.519
6
R5811
59
3.191
16.763
43.037
62.003
77.843.839.397
154.083.204.930.662.557.781.201.849
8
R592.559.647.034.361
4.340.876.285.657.460.212.144.534.289.928.559.826.755.746.751
2
R603
7
11
13
31
37
41
61
101
211
241
271
2.161
3.541
9.091
9.901
27.961
2.906.161
4.188.901
39.526.741
20
R61733
4.637
329.401
974.293
1.360.682.471
106.007.173.861.643
7.061.709.990.156.159.479
7
R6211
2.791
6.943.319
57.336.415.063.790.604.359
909.090.909.090.909.090.909.090.909.091
5
R633
3
37
43
239
1.933
4.649
10.837
23.311
45.613
333.667
10.838.689
45.121.231
1.921.436.048.294.281
14
R6411
17
73
101
137
353
449
641
1.409
19.841
69.857
976.193
5.882.353
6.187.457
834.427.406.578.561
15
R6541
53
79
271
265.371.653
162.503.518.711
5.538.396.997.364.024.056.286.510.640.780.600.481
7
R663
7
11
11
13
23
37
67
4.093
8.779
21.649
513.239
599.144.041
183.411.838.171
1.344.628.210.313.298.373
15
R67493.121
79.863.595.778.924.342.083
28.213.380.943.176.667.001.263.153.660.999.177.245.677
3
R6811
101
103
4.013
2.071.723
28.559.389
1.491.383.821
5.363.222.357
21.993.833.369
2.324.557.465.671.829
10
R693
37
277
203.864.078.068.831
R23
1.595.352.086.329.224.644.348.978.893
6
R7011
41
71
239
271
4.649
9.091
123.551
909.091
4.147.571
102.598.800.232.111.471
265.212.793.249.617.641
12
R71241.573.142.393.627.673.576.957.439.049
45.994.811.347.886.846.310.221.728.895.223.034.301.839
2
R723
3
7
11
13
19
37
73
101
137
3.169
9.901
52.579
98.641
333.667
99.990.001
999.999.000.001
3.199.044.596.370.769
18
R7312.171.337.159
1.855.193.842.151.350.117
49.207.341.634.646.326.934.001.739.482.502.131.487.446.637
3
R7411
7.253
2.028.119
247.629.013
422.650.073.734.453
296.557.347.313.446.299
2.212.394.296.770.203.368.013
7
R753
31
37
41
151
271
4.201
21.401
25.601
2.906.161
182.521.213.001
15.763.985.553.739.191.709.164.170.940.063.151
12
R7611
101
722.817.036.322.379.041
909.090.909.090.909.091
R19
1.369.778.187.490.592.461
6
R77239
4.649
5.237
21.649
42.043
513.239
29.920.507
136.614.668.576.002.329.371.496.447.555.915.740.910.181.043
8
R783
7
11
13
13
37
53
79
157
859
6.397
216.451
265.371.653
1.058.313.049
388.847.808.493
900.900.900.900.990.990.990.991
16
R79317
6.163
10.271
307.627
49.172.195.536.083.790.769
3.660.574.762.725.521.461.527.140.564.875.080.461.079.917
6
R8011
17
41
73
101
137
271
3.541
9.091
27.961
1.676.321
5.070.721
5.882.353
5.964.848.081
19.721.061.166.646.717.498.359.681
15
R813
3
3
3
37
163
757
9.397
333.667
2.462.401
440.334.654.777.631
676.421.558.270.641
130.654.897.808.007.778.425.046.117
13
R8211
83
1.231
538.987
2.670.502.781.396.266.997
3.404.193.829.806.058.997.303
201.763.709.900.322.803.748.657.942.361
7
R833.367.147.378.267
9.512.538.508.624.154.373.682.136.329
346.895.716.385.857.804.544.741.137.394.505.425.384.477
3
R843
7
7
11
13
29
37
43
101
127
239
281
1.933
2.689
4.649
9.901
226.549
459.691
909.091
10.838.689
121.499.449
4.458.192.223.320.340.849
22
R8541
271
2.071.723
262.533.041
5.363.222.357
8.119.594.779.271
4.222.100.119.405.530.170.179.331.190.291.488.789.678.081
7
R8611
173
1.527.791
57.009.401
2.182.600.451
1.963.506.722.254.397
2.140.992.015.395.526.641
7.306.116.556.571.817.748.755.241
8
R873
37
3.191
4.003
16.763
43.037
62.003
72.559
77.843.839.397
310.170.251.658.029.759.045.157.793.237.339.498.342.763.245.483
10
R8811
11
23
73
89
101
137
617
4.093
8.779
21.649
513.239
1.052.788.969
1.056.689.261
16.205.834.846.012.967.584.927.082.656.402.106.953
15
R89497.867
103.733.951
104.984.505.733
5.078.554.966.026.315.671.444.089
403.513.310.222.809.053.284.932.818.475.878.953.159
5
R903
3
7
11
13
19
31
37
41
211
241
271
2.161
9.091
29.611
52.579
238.681
333.667
2.906.161
3.762.091
8.985.695.684.401
4.185.502.830.133.110.721
22
R9153
79
239
547
4.649
14.197
17.837
4.262.077
265.371.653
43.442.141.653
316.877.365.766.624.209
110.742.186.470.530.054.291.318.013
12
R9211
47
101
139
1.289
2.531
18.371.524.594.609
549.797.184.491.917
R23
4.181.003.300.071.669.867.932.658.901
10
R933
37
2.791
6.943.319
57.336.415.063.790.604.359
900.900.900.900.900.900.900.900.900.900.990.990.990.990.990.990.990.990.990.991
6
R9411
6.299
35.121.409
4.855.067.598.095.567
297.262.705.009.139.006.771.611.927
316.362.908.763.458.525.001.406.154.038.726.382.279
6
R9541
191
271
59.281
63.841
R19
1.289.981.231.950.849.543.985.493.631
965.194.617.121.640.791.456.070.347.951.751
8
R963
7
11
13
17
37
73
97
101
137
353
449
641
1.409
9.901
69.857
206.209
5.882.353
99.990.001
66.554.101.249
75.118.313.082.913
9.999.999.900.000.001
22
R9712.004.721
846.035.731.396.919.233.767.211.537.899.097.169
109.399.846.855.370.537.540.339.266.842.070.119.107.662.296.580.348.039
3
R9811
197
239
4.649
909.091
505.885.997
1.976.730.144.598.190.963.568.023.014.679.333
5.076.141.624.365.532.994.918.781.726.395.939.035.533
8
R993
3
37
67
199
397
21.649
34.849
333.667
513.239
1.344.628.210.313.298.373
362.853.724.342.990.469.324.766.235.474.268.869.786.311.886.053.883
12
R10011
41
101
251
271
3.541
5.051
9.091
21.401
25.601
27.961
60.101
7.019.801
182.521.213.001
14.103.673.319.201
78.875.943.472.201
1.680.588.011.350.901
17

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