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Lucas–Carmichael number

Type of positive composite integer

Type of positive composite integer

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.

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 N(X) denote the number of Lucas–Carmichael numbers up to X, Wright showed that there exists a positive constant K such that

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

List of Lucas–Carmichael numbers

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

9868715= 5 × 43 × 197 × 233

References

References

  1. Thomas Wright. (2018). "There are infinitely many elliptic Carmichael numbers". [[Bull. London Math. Soc.]].
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