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 (since n³ + 1 = (n + 1)(n² − n + 1) is always divisible by n + 1).

They are named after Édouard Lucas and Robert Carmichael.

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.^[1] If we let ${\displaystyle N(X)}$ denote the number of Lucas–Carmichael numbers up to ${\displaystyle X}$, Wright showed that there exists a positive constant ${\displaystyle K}$ such that

${\displaystyle N(X)\gg X^{K/\left(\log \log \log X\right)^{2}}}$.

The first few Lucas–Carmichael numbers (sequence A006972 in the OEIS) and their prime factors are listed below.

• Richard Guy (2004). "Section A13". Unsolved Problems in Number Theory (3rd ed.). Springer Verlag.
• Lucas–Carmichael number at PlanetMath.
• "Something special about 399 (and 2015) - Numberphile". YouTube. 15 January 2015. Archived from the original on 2021-12-22.