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 2 to 3pm.

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

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

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