In mathematics, a Lucas–Carmichael number is a positive composite integer n such that If p is a prime factor of n, then p + 1 is a factor of n + 1; 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 first few Lucas–Carmichael numbers are: 399, 935, 2015, 2915, 4991, 5719, 7055, 8855, 12719, 18095, 20705...
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
.
See also Carmichael number Lucas number Lucas pseudoprime Lucas sequence
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