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

Abstract

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) .

Collisions of digit sums in bases 2 and 3

“…This small remark is in fact the main idea that started the research on the present paper. By our key property (41) it is sufficient to prove the existence of (many…”

“…We are going to adapt the Mauduit-Rivat method for digital problems [14,15,16,18,29,30,31,32,34,35,36,37], also applied in the papers [17,41,42,43,44,53]. This will be used in order to obtain a statement concerning the number of n such that f (n) ∈ mZ, where n varies in an arithmetic progression.…”