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 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
Some 'NMBRTHRY@LISTSERV.NODAK.EDU' messages about repunits
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".
https://mersenneforum.org/showthread.php?p=578120#
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) ]
see also my table at webpage about circular primes.References
Serge Batalov “New repunit (PRP) prime found, 5794777 decimal digits, a PRP record”.
https://www.mersenneforum.org/showthread.php?p=576330#
https://groups.io/g/primenumberstheory/message/94
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)
see also my table at webpage about circular primes.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
| Repunit | Prime factors | Comments |
|---|---|---|
| R1 | 1 | 1 trivial case |
| R2 | Is Prime! | 1 |
| R3 | 3 37 | 2 |
| R4 | 11 101 | 2 |
| R5 | 41 271 | 2 |
| R6 | 3 7 11 13 37 | 5 |
| R7 | 239 4.649 | 2 |
| R8 | 11 73 101 137 | 4 |
| R9 | 3 3 37 333.667 | 4 |
| R10 | 11 41 271 9.091 | 4 |
| R11 | 21.649 513.239 | 2 |
| R12 | 3 7 11 13 37 101 9.901 | 7 |
| R13 | 53 79 265.371.653 | 3 |
| R14 | 11 239 4.649 909.091 | 4 |
| R15 | 3 31 37 41 271 2.906.161 | 6 |
| R16 | 11 17 73 101 137 5.882.353 | 6 |
| R17 | 2.071.723 5.363.222.357 | 2 |
| R18 | 3 3 7 11 13 19 37 52.579 333.667 | 9 |
| R19 | Is Prime! | 1 |
| R20 | 11 41 101 271 3.541 9.091 27.961 | 7 |
| R21 | 3 37 43 239 1.933 4.649 10.838.689 | 7 |
| R22 | 11 11 23 4.093 8.779 21.649 513.239 | 7 |
| R23 | Is Prime! | 1 |
| R24 | 3 7 11 13 37 73 101 137 9.901 99.990.001 | 10 |
| R25 | 41 271 21.401 25.601 182.521.213.001 | 5 |
| R26 | 11 53 79 859 265.371.653 1.058.313.049 | 6 |
| R27 | 3 3 3 37 757 333.667 440.334.654.777.631 | 7 |
| R28 | 11 29 101 239 281 4.649 909.091 121.499.449 | 8 |
| R29 | 3.191 16.763 43.037 62.003 77.843.839.397 | 5 |
| R30 | 3 7 11 13 31 37 41 211 241 271 2.161 9.091 2.906.161 | 13 |
| R31 | 2.791 6.943.319 57.336.415.063.790.604.359 | 3 |
| R32 | 11 17 73 101 137 353 449 641 1.409 69.857 5.882.353 | 11 |
| R33 | 3 37 67 21.649 513.239 1.344.628.210.313.298.373 | 6 |
| R34 | 11 103 4.013 2.071.723 5.363.222.357 21.993.833.369 | 6 |
| R35 | 41 71 239 271 4.649 123.551 102.598.800.232.111.471 | 7 |
| R36 | 3 3 7 11 13 19 37 101 9.901 52.579 333.667 999.999.000.001 | 12 |
| R37 | 2.028.119 247.629.013 2.212.394.296.770.203.368.013 | 3 |
| R38 | 11 909.090.909.090.909.091 R19 | 3 |
| R39 | 3 37 53 79 265.371.653 900.900.900.900.990.990.990.991 | 6 |
| R40 | 11 41 73 101 137 271 3.541 9.091 27.961 1.676.321 5.964.848.081 | 11 |
| R41 | 83 1.231 538.987 201.763.709.900.322.803.748.657.942.361 | 4 |
| R42 | 3 7 7 11 13 37 43 127 239 1.933 2.689 4.649 459.691 909.091 10.838.689 | 15 |
| R43 | 173 1.527.791 1.963.506.722.254.397 2.140.992.015.395.526.641 | 4 |
| R44 | 11 11 23 89 101 4.093 8.779 21.649 513.239 1.052.788.969 1.056.689.261 | 11 |
| R45 | 3 3 31 37 41 271 238.681 333.667 2.906.161 4.185.502.830.133.110.721 | 10 |
| R46 | 11 47 139 2.531 549.797.184.491.917 R23 | 6 |
| R47 | 35.121.409 316.362.908.763.458.525.001.406.154.038.726.382.279 | 2 |
| R48 | 3 7 11 13 17 37 73 101 137 9.901 5.882.353 99.990.001 9.999.999.900.000.001 | 13 |
| R49 | 239 4.649 505.885.997 1.976.730.144.598.190.963.568.023.014.679.333 | 4 |
| R50 | 11 41 251 271 5.051 9.091 21.401 25.601 182.521.213.001 78.875.943.472.201 | 10 |
| R51 | 3 37 613 210.631 2.071.723 52.986.961 5.363.222.357 13.168.164.561.429.877 | 8 |
| R52 | 11 53 79 101 521 859 265.371.653 1.058.313.049 1.900.381.976.777.332.243.781 | 9 |
| R53 | 107 1.659.431 1.325.815.267.337.711.173 47.198.858.799.491.425.660.200.071 | 4 |
| R54 | 3 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 |
| R55 | 41 271 1.321 21.649 62.921 513.239 83.251.631 1.300.635.692.678.058.358.830.121 | 8 |
| R56 | 11 29 73 101 137 239 281 4.649 7.841 909.091 121.499.449 127.522.001.020.150.503.761 | 12 |
| R57 | 3 37 21.319 10.749.631 R19 3.931.123.022.305.129.377.976.519 | 6 |
| R58 | 11 59 3.191 16.763 43.037 62.003 77.843.839.397 154.083.204.930.662.557.781.201.849 | 8 |
| R59 | 2.559.647.034.361 4.340.876.285.657.460.212.144.534.289.928.559.826.755.746.751 | 2 |
| R60 | 3 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 |
| R61 | 733 4.637 329.401 974.293 1.360.682.471 106.007.173.861.643 7.061.709.990.156.159.479 | 7 |
| R62 | 11 2.791 6.943.319 57.336.415.063.790.604.359 909.090.909.090.909.090.909.090.909.091 | 5 |
| R63 | 3 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 |
| R64 | 11 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 |
| R65 | 41 53 79 271 265.371.653 162.503.518.711 5.538.396.997.364.024.056.286.510.640.780.600.481 | 7 |
| R66 | 3 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 |
| R67 | 493.121 79.863.595.778.924.342.083 28.213.380.943.176.667.001.263.153.660.999.177.245.677 | 3 |
| R68 | 11 101 103 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 |
| R69 | 3 37 277 203.864.078.068.831 R23 1.595.352.086.329.224.644.348.978.893 | 6 |
| R70 | 11 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 |
| R71 | 241.573.142.393.627.673.576.957.439.049 45.994.811.347.886.846.310.221.728.895.223.034.301.839 | 2 |
| R72 | 3 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 |
| R73 | 12.171.337.159 1.855.193.842.151.350.117 49.207.341.634.646.326.934.001.739.482.502.131.487.446.637 | 3 |
| R74 | 11 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 |
| R75 | 3 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 |
| R76 | 11 101 722.817.036.322.379.041 909.090.909.090.909.091 R19 1.369.778.187.490.592.461 | 6 |
| R77 | 239 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 |
| R78 | 3 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 |
| R79 | 317 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 |
| R80 | 11 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 |
| R81 | 3 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 |
| R82 | 11 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 |
| R83 | 3.367.147.378.267 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 |
| R84 | 3 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 |
| R85 | 41 271 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 |
| R86 | 11 173 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 |
| R87 | 3 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 |
| R88 | 11 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 |
| R89 | 497.867 103.733.951 104.984.505.733 5.078.554.966.026.315.671.444.089 403.513.310.222.809.053.284.932.818.475.878.953.159 | 5 |
| R90 | 3 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 |
| R91 | 53 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 |
| R92 | 11 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 |
| R93 | 3 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 |
| R94 | 11 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 |
| R95 | 41 191 271 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 |
| R96 | 3 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 |
| R97 | 12.004.721 846.035.731.396.919.233.767.211.537.899.097.169 109.399.846.855.370.537.540.339.266.842.070.119.107.662.296.580.348.039 | 3 |
| R98 | 11 197 239 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 |
| R99 | 3 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 |
| R100 | 11 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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Patrick De Geest - Belgium
- Short Bio - Some PicturesE-mail address : pdg@worldofnumbers.com