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Author: search.filters.author.Kim, VeasnaHas files: No

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    On r-free integers in Beatty sequences
    (2022-07-01)
    Kim, Veasna
    ;
    Srichan, Teerapat
    ;
    Mavecha, Sukrawan
    Let r≥ 2 be a fixed integer. A positive integer n is called r-free if in the canonical representation of n into prime powers each exponent is < r. The integer 1 is considered to be r-free. In this paper, we consider Q<inf>r</inf>(x; α, β) , which is the number of r-free integers of Beatty sequence ⌊ αn+ β⌋ , 1 ≤ n≤ x, for α> 1 irrational and with bounded partial quotients, β∈ [0 , α). We prove that, as x→ ∞Qr(x;α,β)=xζ(r)+O(x(r+1)/2rlog3x),which improves Victorovich’s result in the case of square-free integers. Moreover, we also prove there exist infinitely many consecutive square-free numbers of the forms ⌊ αn+ β⌋ , ⌊ αn+ β⌋ + 1 , which improves Dimitrov’s result in 2019.
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    A divided-difference characterization of polynomials over finite fields of characteristic two
    (2022-04-01)
    Kim, Veasna
    ;
    Laohakosol, Vichian
    ;
    Mavecha, Sukrawan
    For a finite field F of characteristic 2, an integer n≥ 3 , and for a function f: F→ F, if there is a function h: F→ F such that the divided difference f[x<inf>1</inf>, ⋯ , x<inf>n</inf>] on any n distinct elements of F satisfies f[x<inf>1</inf>, … , x<inf>n</inf>] = h(x<inf>1</inf>+ ⋯ + x<inf>n</inf>) , then f is a polynomial of degree at most n over F. This makes earlier work of Davies and Rousseau complete.
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