Lucas–Carmichael number
In mathematics, a Lucas–Carmichael number is a positive composite integer n such that
- if p is a prime factor of n, then p + 1 is a factor of n + 1;
- 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
[edit]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 denote the number of Lucas–Carmichael numbers up to , Wright showed that there exists a positive constant such that
.
List of Lucas–Carmichael numbers
[edit]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
[edit]- ^ 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.
External links
[edit]- 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.