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\title{A Formalisation of Lehmer's Primality Criterion}
\author{By Simon Wimmer and Lars Noschinski}
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\begin{abstract}
  In 1927, Lehmer presented criterions for primality, based on the
  converse of Fermat's litte theorem~\cite{lehmer1927fermat_converse}.
  This work formalizes the second criterion from Lehmer's paper,
  a necessary and sufficient condition for primality.

  As a side product we formalize some properties of Euler's $\varphi$-function,
  the notion of the order of an element of a group, and the cyclicity of the
  multiplicative group of a finite field.
\end{abstract}

\tableofcontents

\section{Introduction}

Section \ref{sec:simp-rules} provides some technical lemmas about polynomials.
Section \ref{sec:euler-phi} to \ref{sec:number-roots} formalize some basic number-theoretic
and algebraic properties: Euler's $\varphi$-function, the order of an element of a group
and an upper bound of the number of roots of a polynomial. Section \ref{sec:mult-group}
combines these results to prove that the multiplicative group of a finite field is cyclic.
Based on that, Section \ref{sec:lehmer} formalizes an extended version of Lehmer's Theorem,
which gives us necessary and sufficient conditions to decide whether a number is prime.

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