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### Landau’s function for one million billions

Journal de Théorie des Nombres de Bordeaux

Let ${𝔖}_{n}$ denote the symmetric group with $n$ letters, and $g\left(n\right)$ the maximal order of an element of ${𝔖}_{n}$. If the standard factorization of $M$ into primes is $M={q}_{1}^{{\alpha }_{1}}{q}_{2}^{{\alpha }_{2}}...{q}_{k}^{{\alpha }_{k}}$, we define $\ell \left(M\right)$ to be ${q}_{1}^{{\alpha }_{1}}+{q}_{2}^{{\alpha }_{2}}+...+{q}_{k}^{{\alpha }_{k}}$; one century ago, E. Landau proved that $g\left(n\right)={max}_{\ell \left(M\right)\le n}M$ and that, when $n$ goes to infinity, $logg\left(n\right)\sim \sqrt{nlog\left(n\right)}$. There exists a basic algorithm to compute $g\left(n\right)$ for $1\le n\le N$; its running time is $𝒪\left({N}^{3/2}/\sqrt{logN}\right)$ and the needed memory is $𝒪\left(N\right)$; it allows computing $g\left(n\right)$ up to, say, one million. We describe an algorithm to calculate $g\left(n\right)$ for $n$ up to ${10}^{15}$. The main idea is to use the so-called ...

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