Lucas–Carmichael number


In mathematics, a Lucas–Carmichael number is a positive composite integer n such that
  1. If p is a prime factor of n, then p + 1 is a factor of n + 1;
  2. n is odd and square-free.
The first condition resembles Korselt's criterion for Carmichael numbers, where −1 is replaced with +1. The second condition eliminates from consideration some trivial cases like cubes of prime numbers, such as 8 or 27, which otherwise would be Lucas–Carmichael numbers.
They are named after Édouard Lucas and Robert Carmichael.

Properties

The smallest Lucas–Carmichael number is 399 = 3 × 7 × 19. It is easy to verify that 3+1, 7+1, and 19+1 are all factors of 399+1 = 400.
The smallest Lucas–Carmichael number with 4 factors is 8855 = 5 × 7 × 11 × 23.
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.
Thomas Wright proved in 2016 that there are infinitely many Lucas–Carmichael numbers. If we let denote the number of Lucas–Carmichael numbers up to, Wright showed that there exists a positive constant such that

List of Lucas–Carmichael numbers

The first few Lucas–Carmichael numbers and their prime factors are listed below.