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.
By convention, a number is only regarded as a Lucas–Carmichael number if it is odd and square-free (not divisible by the square of a prime number), otherwise any cube of a prime number, such as 8 or 27, would be a Lucas–Carmichael number (since n3 + 1 = (n + 1)(n2 − n + 1) is always divisible by n + 1).
Thus the smallest such number is 399 = 3 × 7 × 19; 399+1 = 400; 3+1, 7+1 and 19+1 are all factors of 400.
The first few numbers, and their factors, are (sequence A006972 in the OEIS):
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 × 291 |
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 |
The smallest LucasCarmichael 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.
References
- Unsolved Problems in Number Theory (3rd edition) by Richard Guy (Springer Verlag, 2004), section A13.
- PlanetMath