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# 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 &times; 7 &times; 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 &times; 7 &times; 19

:935 = 5 &times; 11 &times; 17

:2015 = 5 &times; 13 &times; 31

:2915 = 5 &times; 11 &times; 53

:4991 = 7 &times; 23 &times; 31

:5719 = 7 &times; 19 &times; 43

:7055 = 5 &times; 17 &times; 83

:8855 = 5 &times; 7 &times; 11 &times; 23

:12719 = 7 &times; 23 &times; 79

:18095 = 5 &times; 7 &times; 11 &times; 47

:20999 = 11 &times; 23 &times; 83

:22847 = 11 &times; 31 &times; 67

:29315 = 5 &times; 11 &times; 13 &times; 41

:31535 = 5 &times; 7 &times; 17 &times; 53

:46079 = 11 &times; 59 &times; 71

:51359 = 7 &times; 11 &times; 23 &times; 291

:76751 = 23 &times; 47 &times; 71

:80189 = 17 &times; 53 &times; 89

:81719 = 11 &times; 17 &times; 19 &times; 23

:88559 = 19 &times; 59 &times; 79

:104663 = 13 &times; 83 &times; 97

The smallest Lucas-Carmichael number with 5 factors is 588455 = 5 &times; 7 &times; 17 &times; 23 &times; 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]

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