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 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