On a problem of Chen and Liu concerning the prime power factorization of $n!$

Sur un problème de Chen et Liu concernant la factorisation en puissances des premiers de $n!$

Thomas Stoll (1) , Johannes F. Morgenbesser (2)

1 AMU - Aix Marseille Université
2 Fakultät für Mathematik [Wien]

For a fixed prime p, let e_p(n!) denote the order of p in the prime factorization of n!. Chen and Liu (2007) asked whether for any fixed m, one has {e_p(n^2 !) mod m : n ∈ Z} = Z_m and {e_p(q!) mod m : q prime} = Z_m. We answer these two questions and show asymptotic formulas for #{n < x : n ≡ a mod d, e_p (n^2 !) ≡ r mod m} and #{q < x : q prime, q ≡ a mod d, e_p (q!) ≡ r mod m}. Furthermore, we show that for each h ≥ 3, we have {n < x : n ≡ a mod d, e_p (n^h !) ≡ r mod m} >>x^(4/(3h+1)).

Mots clés: Secondary 11A63; prime power factorization; p-adic valuation; sum of digits; congruences; squares; primes

Domaines: Théorie des nombres [math.NT]

Thomas Stoll, Johannes F. Morgenbesser. On a problem of Chen and Liu concerning the prime power factorization of $n!$. Proceedings of the American Mathematical Society, 2013, 141 (7), pp.2289-2297. ⟨10.1090/S0002-9939-2013-11751-X⟩. ⟨hal-01278662⟩