Lucas-Carmichael number
In mathematics, a Lucas-Carmichael number is a positive integer n such that if p is a prime factor of n, then p + 1 is a factor of n + 1. By custom, 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. 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 : - 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
- 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
- 76751 = 23 × 47 × 71
- 80189 = 17 × 53 × 89
- 81719 = 11 × 17 × 19 × 23
- 88559 = 19 × 59 × 79
- 104663 = 13 × 83 × 97
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.
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