Lucas–Carmichael number

In mathematics, a Lucas–Carmichael number is a positive composite integer n such that

  1. if p is a prime factor of n, then p + 1 is a factor of n + 1;
  2. n is odd and square-free.

The first condition resembles the Korselt's criterion for Carmichael numbers, where -1 is replaced with +1. The second condition eliminates from consideration some trivial cases like cubes of prime numbers, such as 8 or 27, which otherwise would be Lucas–Carmichael numbers (since n3 + 1 = (n + 1)(n2 − n + 1) is always divisible by n + 1).

They are named after Édouard Lucas and Robert Carmichael.

Properties

The smallest Lucas–Carmichael number is 399 = 3 × 7 × 19. It is easy to verify that 3+1, 7+1, and 19+1 are all factors of 399+1 = 400.

The smallest Lucas–Carmichael number with 4 factors is 8855 = 5 × 7 × 11 × 23.

The smallest Lucas–Carmichael number with 5 factors is 588455 = 5 × 7 × 17 × 23 × 43.

It is not known whether any Lucas–Carmichael number is also a Carmichael number.

List of Lucas–Carmichael numbers

The first few Lucas–Carmichael numbers (sequence A006972 in the OEIS) and their prime factors are listed below.

399= 3 × 7 × 19
935= 5 × 11 × 17
2015= 5 × 13 × 31
2915= 5 × 11 × 53
4991= 7 × 23 × 31
5719= 7 × 19 × 43
7055= 5 × 17 × 83
8855= 5 × 7 × 11 × 23
12719= 7 × 23 × 79
18095= 5 × 7 × 11 × 47
20705= 5 × 41 × 101
20999= 11 × 23 × 83
22847= 11 × 31 × 67
29315= 5 × 11 × 13 × 41
31535= 5 × 7 × 17 × 53
46079= 11 × 59 × 71
51359= 7 × 11 × 23 × 29
60059= 19 × 29 × 109
63503= 11 × 23 × 251
67199= 11 × 41 × 149
73535= 5 × 7 × 11 × 191
76751= 23 × 47 × 71
80189= 17 × 53 × 89
81719= 11 × 17 × 19 × 23
88559= 19 × 59 × 79
90287= 17 × 47 × 113
104663= 13 × 83 × 97
117215= 5 × 7 × 17 × 197
120581= 17 × 41 × 173
147455= 5 × 7 × 11 × 383
152279= 29 × 59 × 89
155819= 19 × 59 × 139
162687= 3 × 7 × 61 × 127
191807= 7 × 11 × 47 × 53
194327= 7 × 17 × 23 × 71
196559= 11 × 107 × 167
214199= 23 × 67 × 139
218735= 5 × 11 × 41 × 97
230159= 47 × 59 × 83
265895= 5 × 7 × 71 × 107
357599= 11 × 19 × 29 × 59
388079= 23 × 47 × 359
390335= 5 × 11 × 47 × 151
482143= 31 × 103 × 151
588455= 5 × 7 × 17 × 23 × 43
653939= 11 × 13 × 17 × 269
663679= 31 × 79 × 271
676799= 19 × 179 × 199
709019= 17 × 179 × 233
741311= 53 × 71 × 197
760655= 5 × 7 × 103 × 211
761039= 17 × 89 × 503
776567= 11 × 227 × 311
798215= 5 × 11 × 23 × 631
880319= 11 × 191 × 419
895679= 17 × 19 × 47 × 59
913031= 7 × 23 × 53 × 107
966239= 31 × 71 × 439
966779= 11 × 179 × 491
973559= 29 × 59 × 569
1010735= 5 × 11 × 17 × 23 × 47
1017359= 7 × 23 × 71 × 89
1097459= 11 × 19 × 59 × 89
1162349= 29 × 149 × 269
1241099= 19 × 83 × 787
1256759= 7 × 17 × 59 × 179
1525499= 53 × 107 × 269
1554119= 7 × 53 × 59 × 71
1584599= 37 × 113 × 379
1587599= 13 × 97 × 1259
1659119= 7 × 11 × 29 × 743
1707839= 7 × 29 × 47 × 179
1710863= 7 × 11 × 17 × 1307
1719119= 47 × 79 × 463
1811687= 23 × 227 × 347
1901735= 5 × 11 × 71 × 487
1915199= 11 × 13 × 59 × 227
1965599= 79 × 139 × 179
2048255= 5 × 11 × 167 × 223
2055095= 5 × 7 × 71 × 827
2150819= 11 × 19 × 41 × 251
2193119= 17 × 23 × 71 × 79
2249999= 19 × 79 × 1499
2276351= 7 × 11 × 17 × 37 × 47
2416679= 23 × 179 × 587
2581319= 13 × 29 × 41 × 167
2647679= 31 × 223 × 383
2756159= 7 × 17 × 19 × 23 × 53
2924099= 29 × 59 × 1709
3106799= 29 × 149 × 719
3228119= 19 × 23 × 83 × 89
3235967= 7 × 17 × 71 × 383
3332999= 19 × 23 × 29 × 263
3354695= 5 × 17 × 61 × 647
3419999= 11 × 29 × 71 × 151
3441239= 109 × 131 × 241
3479111= 83 × 167 × 251
3483479= 19 × 139 × 1319
3700619= 13 × 41 × 53 × 131
3704399= 47 × 269 × 293
3741479= 7 × 17 × 23 × 1367
4107455= 5 × 11 × 17 × 23 × 191
4285439= 89 × 179 × 269
4452839= 37 × 151 × 797
4587839= 53 × 107 × 809
4681247= 47 × 103 × 967
4853759= 19 × 23 × 29 × 383
4874639= 7 × 11 × 29 × 37 × 59
5058719= 59 × 179 × 479
5455799= 29 × 419 × 449
5669279= 7 × 11 × 17 × 61 × 71
5807759= 83 × 167 × 419
6023039= 11 × 29 × 79 × 239
6514199= 43 × 197 × 769
6539819= 11 × 13 × 19 × 29 × 83
6656399= 29 × 89 × 2579
6730559= 11 × 23 × 37 × 719
6959699= 59 × 179 × 659
6994259= 17 × 467 × 881
7110179= 37 × 41 × 43 × 109
7127999= 23 × 479 × 647
7234163= 17 × 41 × 97 × 107
7274249= 17 × 449 × 953
7366463= 13 × 23 × 71 × 347
8159759= 19 × 29 × 59 × 251
8164079= 7 × 11 × 229 × 463
8421335= 5 × 13 × 23 × 43 × 131
8699459= 43 × 307 × 659
8734109= 37 × 113 × 2089
9224279= 53 × 269 × 647
9349919= 19 × 29 × 71 × 239
9486399= 3 × 13 × 79 × 3079
9572639= 29 × 41 × 83 × 97
9694079= 47 × 239 × 863
9868715= 5 × 43 × 197 × 233
  • Richard Guy (2004). "Section A13". Unsolved Problems in Number Theory (3rd ed.). Springer Verlag.
  • Lucas–Carmichael number at PlanetMath.
  • "Something special about 399 (and 2015) - Numberphile". YouTube. Archived from the original on 2021-12-22.
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