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 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 $N(X)$ denote the number of Lucas–Carmichael numbers up to $X$ , Wright showed that there exists a positive constant $K$ such that

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

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

^ 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

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. 15 January 2015. Archived from the original on 2021-12-22.

[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.

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