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 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 OEIS|id=A006972:

: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.

References

* "Unsolved Problems in Number Theory" (3rd edition) by Richard Guy (Springer Verlag, 2004), section A13.
* [http://planetmath.org/encyclopedia/LucasCarmichaelNumber.html PlanetMath]