# Lucas–Carmichael number

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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.

Thomas Wright proved in 2016 that there are infinitely many Lucas–Carmichael numbers.[1] If we let ${\displaystyle N(X)}$ denote the number of Lucas–Carmichael numbers up to ${\displaystyle X}$, Wright showed that there exists a positive constant ${\displaystyle K}$ such that

${\displaystyle N(X)\gg X^{K/\left(\log \log \log X\right)^{2}}}$.

## 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

## References

1. ^ Thomas Wright (2018). "There are infinitely many elliptic Carmichael numbers". Bull. London Math. Soc. 50 (5): 791–800. arXiv:1609.00231. doi:10.1112/blms.12185. S2CID 119676706.