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Landau’s function for one million billions

Marc DelégliseJean-Louis NicolasPaul Zimmermann — 2008

Journal de Théorie des Nombres de Bordeaux

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

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