Presenter: Max Lewis (UQ)

A Carmichael number is a composite number $n$ that satisfies $a^{n-1} \equiv 1 \mod{n}$, for all integers $a$ such that $\gcd(a,\, n) = 1$. They are of interest in primality testing. Hsu defines Carmichael polynomials in $\mathbb{F}_q[x]$ for any prime power $q$, and shows that there are infinitely many such polynomials. We prove that for any $\mathbf{a},\, \mathbf{M} \in \mathbb{F}_q[x]$ where $\mathbf{M}$ is monic and $\gcd(\mathbf{a},\, \mathbf{M}) = 1$, there are infinitely many Carmichael polynomials that are congruent to $\mathbf{a}$ modulo $\mathbf{M}$. We also define analogues of Grau and Oller-Marc\'{e}n's $k$-Lehmer numbers and prove that for infinitely many $k$ there are Carmichael polynomials that are $k$-Lehmer but not $(k-1)$-Lehmer.

### About Pure mathematics seminars

We present regular seminars on a range of pure mathematics interests. Students, staff and visitors to UQ are welcome to attend, and to suggest speakers and topics.

Seminars are usually held on Tuesdays from 3pm to 3.50pm.

Talks comprise 45 minutes of speaking time plus five minutes for questions and discussion.

#### Information for speakers

Researchers in all pure mathematics fields attend our seminars, so please aim your presentation at a general mathematical audience.

#### Contact us

To volunteer to talk or to suggest a speaker, email Ole Warnaar or Travis Scrimshaw.