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Coprimality of Consecutive Elements in a Piatetski-Shapiro Sequence

In: Number Theory in Memory of Eduard Wirsing

Author

Listed:
  • Jean-Marc Deshouillers

    (Université de Bordeaux, Bordeaux INP, CNRS, Institut de Mathématiques de Bordeaux, UMR 5251)

  • Michael Drmota

    (TU Wien, Institute of Discrete Mathematics and Geometry)

  • Clemens Müllner

    (TU Wien, Institute of Discrete Mathematics and Geometry)

Abstract

We show that in any Piatetski-Shapiro sequence ⌊ n c ⌋ n $$\left (\lfloor n^c \rfloor \right )_n$$ with c in ( 1 , + ∞ ) ∖ ℕ $$(1, +\infty )\backslash \mathbb {N}$$ , there exist long subsequences of consecutive elements no pair of which are coprime, whereas for any c in ( 1 , 2 ) $$(1, 2)$$ , there exist infinitely many n such that all the elements in { ⌊ n c ⌋ , ⌊ ( n + 1 ) c ⌋ , … , ⌊ ( n + H ) c ⌋ } $$\{\lfloor n^c \rfloor , \lfloor (n+1)^c \rfloor , \ldots , \lfloor (n+H)^c \rfloor \}$$ are pairwise coprime for H almost as large as min ( c − 1 , 1 − c ∕ 2 ) log n $$\min (c-1,1-c/2)\log n$$ .

Suggested Citation

  • Jean-Marc Deshouillers & Michael Drmota & Clemens Müllner, 2023. "Coprimality of Consecutive Elements in a Piatetski-Shapiro Sequence," Springer Books, in: Helmut Maier & Jörn Steuding & Rasa Steuding (ed.), Number Theory in Memory of Eduard Wirsing, pages 91-98, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-31617-3_7
    DOI: 10.1007/978-3-031-31617-3_7
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