Polynomial products modulo primes and applications. (English) Zbl 1446.11030
For a polynomial \(P(x)\in{\mathbb Z}[x]\) and \(n\geq 1\), put \(F_P(n)=\prod_{k\leq n} P(k)\pmod{p}\), where \(p\) is a prime. Perfect power values of this expression are investigated by many authors; see e.g. results of Erdős-Selfridge (linear case) or Cilleruelo (quadratic case).
The authors study arithmetic dynamical systems generated by \(F_P(n)\), and provide several related results. Among others, they show that
\[ \# S_d(M,N)\ll N^{7/8}(\log N)^{1/4}, \]
where \(S_d(M,N)\) \((d,M,N\geq 1)\) is the set of integers \(n\), for which all the numbers \(n=M+1,\ldots,M+N\) can be written in the form \(dt^2\). (This result improves an estimate of Cilleruelo, Luca, Quirós and Shparlinski.) As an application, they give a new, improved lower bound on the number of distinct quadratic fields of the form \(\mathbb Q(\sqrt{F_P(n)})\) in short intervals. They give further related results, as well, for example, for the number of missing values of \(F_P(n)\pmod{p}\) for certain special families of polynomials, generalizing several theorems from the literature.
The authors study arithmetic dynamical systems generated by \(F_P(n)\), and provide several related results. Among others, they show that
\[ \# S_d(M,N)\ll N^{7/8}(\log N)^{1/4}, \]
where \(S_d(M,N)\) \((d,M,N\geq 1)\) is the set of integers \(n\), for which all the numbers \(n=M+1,\ldots,M+N\) can be written in the form \(dt^2\). (This result improves an estimate of Cilleruelo, Luca, Quirós and Shparlinski.) As an application, they give a new, improved lower bound on the number of distinct quadratic fields of the form \(\mathbb Q(\sqrt{F_P(n)})\) in short intervals. They give further related results, as well, for example, for the number of missing values of \(F_P(n)\pmod{p}\) for certain special families of polynomials, generalizing several theorems from the literature.
Reviewer: Lajos Hajdu (Debrecen)
MSC:
| 11B50 | Sequences (mod \(m\)) |
| 11D45 | Counting solutions of Diophantine equations |
| 11R09 | Polynomials (irreducibility, etc.) |
| 11R11 | Quadratic extensions |
| 11R44 | Distribution of prime ideals |
Keywords:
dynamical system modulo \(p\); distribution of sequences modulo \(p\); Diophantine equations; perfect powers; polynomials; prime ideals of number fieldsReferences:
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