Carmichael and Lehmer numbers and the analogous polynomials

Speaker: Max Lewis
Affiliation: University of Queensland (PhD exit seminar)

Abstract

We define the level of an integer or polynomial n to be the smallest integer k (if it exists) such that n is k-Lehmer. We define k-Lehmer polynomials analogously to to J. M. Grau and A. M. Oller Marcen’s definition of k-Lehmer numbers. In this talk we summarise our results relating to Carmichael numbers and polynomials of level k and the different notion of k-Carmichael numbers. In particular, we show that for infinitely many integers k, there are both Carmichael numbers and polynomials of level k. This follows from a Theorem of T. Wright’s and our extension of it to polynomials. We call the Carmichael depth of an integer n the smallest k such that n is k-Carmichael. We show that for any finite, non-empty set of primes S, there is an integer n whose Carmichael depth is exactly the product of the primes in S. Finally, we strengthen the definition of 1-Lehmer polynomials to create universally Lehmer polynomials. We were unable to find any examples of universally Lehmer polynomials over any finite field (except trivially for the field of two elements). Future work could include examples of such polynomials (or a proof that none exist) and possibly a proof of the infinitude of 1-Lehmer polynomials. We finish with examples of Carmichael polynomials of level k for k > 10^6 that we constructed using a modified subset-product algorithm due to W. R. Alford, J. Grantham, S. Hayman and A. Shallue.