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\title{Mersenne primes and the Lucas--Lehmer test}
\author{Manuel Eberl}
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\begin{abstract}
This article provides formal proofs of basic properties of Mersenne numbers, i.\,e.\ numbers of the form $2^n - 1$, and especially of Mersenne primes.
In particular, an efficient, verified, and executable version of the Lucas--Lehmer test is developed. This test decides primality for Mersenne numbers in
time polynomial in $n$.
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