Editing Lucas–Carmichael number

{{Short description|Type of positive composite integer}}
In [[mathematics]], a '''Lucas–Carmichael number''' is a positive [[Composite number|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 integer|square-free]].
The first condition resembles Korselt's criterion for [[Carmichael number]]s, 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 ''n''<sup>3</sup>&nbsp;+&nbsp;1 =&nbsp;(''n''&nbsp;+&nbsp;1)(''n''<sup>2</sup>&nbsp;&minus;&nbsp;''n''&nbsp;+&nbsp;1) is always divisible by&nbsp;''n''&nbsp;+&nbsp;1).

They are named after [[Édouard Lucas]] and [[Robert Daniel Carmichael|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 (mathematician)|Thomas Wright]] proved in 2016 that there are infinitely many Lucas–Carmichael numbers.<ref>{{cite journal |author=Thomas Wright |title=There are infinitely many elliptic Carmichael numbers |journal=[[Bull. London Math. Soc.]] |volume=50 |year=2018 |issue=5 |pages=791–800 |arxiv=1609.00231 |doi=10.1112/blms.12185 |s2cid=119676706 }}</ref>  If we let <math> N(X)</math> denote the number of Lucas–Carmichael numbers up to <math> X</math>, Wright showed that there exists a positive constant <math>K</math> such that

<math> N(X) \gg X^{K/\left( \log\log \log X\right)^2}</math>.

== List of Lucas–Carmichael numbers ==
The first few Lucas–Carmichael numbers {{OEIS|id=A006972}} and their prime factors are listed below.

{{div col|colwidth=20em}}
{|
|-
| 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
|}
{{div col end}}

==References==
{{Reflist}}

==External links==
* {{cite book |title=Unsolved Problems in Number Theory |edition=3rd |author=Richard Guy |publisher=Springer Verlag |year=2004 |chapter=Section A13}}
* {{PlanetMath|lucascarmichaelnumber}}
* {{cite web |title=Something special about 399 (and 2015) - Numberphile |url=https://www.youtube.com/watch?v=yfr3BIk6KFc  |archive-url=https://ghostarchive.org/varchive/youtube/20211222/yfr3BIk6KFc |archive-date=2021-12-22 |url-status=live|website=YouTube| date=15 January 2015 }}{{cbignore}}

{{Classes of natural numbers}}

{{DEFAULTSORT:Lucas-Carmichael number}}
[[Category:Eponymous numbers in mathematics]]
[[Category:Integer sequences]]