In mathematics, a Lucas–Carmichael number is a positive composite integer n such that The first condition resembles Korselt's criterion for Carmichael numbers, where -1 is replaced with +1.
They are named after Édouard Lucas and Robert Carmichael.
Thomas Wright proved in 2016 that there are infinitely many Lucas–Carmichael numbers.
, Wright showed that there exists a positive constant
The first few Lucas–Carmichael numbers (sequence A006972 in the OEIS) and their prime factors are listed below.