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Patrick Letendre

Publications and source records attributed to Patrick Letendre.

11 recordsLinked to original sources
arXiv2026

The $abc$ Conjecture Revisited

We propose a new abc-type conjecture. We motivate the conjecture and illustrate its relevance through several applications. Our main result concerns the function $$ W(x,y) := \sum_{j = 1}^{y}\omega(x+j) \quad (y \in \mathbb{N},\ x \in \mathbb{Z}_{\ge 0}) $$ where $\omega(n)$ denotes the number of distinct prime divisors of $n$. The new conjecture implies that, for each fixed $y \in \mathbb{N}$, $$ \limsup_{x \to \infty} \frac{W(x,y)\log\log x}{\log x} = 1. $$

math.NT↗
arXiv2025

The Vandermonde Determinant of the Divisors of an Integer

Let $1=d_{1}<d_{2}< \cdots < d_{\tau(n)}=n$ denote the ordered sequence of the positive divisors of an integer $n$. We are interested in estimating the arithmetic function $$ V(n) := \prod_{1 \le i < j \le \tau(n)}(d_{j}-d_{i}) \quad (n \ge 1). $$

math.NT↗
arXiv2025

Divisors of an Integer in a Short Interval

Let $\mathcal{D}_{n} \subset \mathbb{N}$ denote the set of the $\tau(n)$ divisors of $n$. We study the function $$ D_{n}(X,Y):=|\{d \in \mathcal{D}_{n}:\ X \le d \le X+Y\}| $$ for $Y \le X$.

math.NT↗
arXiv2024

Relations in the Set of Divisors of an Integer $n$

Let $\mathcal{D}_{n} \subset \mathbb{N}$ be the set of the $\tau(n)$ divisors of $n$. We generalize a method developed by Erd\H os, Tenenbaum and de la Bret\`eche for the study of the set $\mathcal{D}_{n}$. In particular, using these ideas, we establish that $$ |\{(d_{1},d_{2},d_{3}) \in \mathcal{D}_{n}^3 : d_{1}+d_{2}=d_{3}\}| \le \tau(n)^{2-\delta} $$ with $\delta=0.045072$.

math.NT↗
arXiv2019

Polynomials with integer roots

Let $\mathcal{F}_n$ be the set of unitary polynomials of degree $n \ge 2$ that have their roots in $\mathbb{Z}^*$. We note $$ Q(x) := x^n+a_{1}x^{n-1}+\dots+a_{n}. $$ We show that any two fixed consecutive coefficients $(a_{j},a_{j+1})$ ($j \in \{1,\dots,n-1\})$ define finitely many polynomials of $\mathcal{F}_n$.

math.NT↗
arXiv2019

Subsets of $\mathbb{F}^*_p$ with only small products or ratios

Let $p$ be a fixed prime. We estimate the number of elements of a set $A \subseteq \mathbb{F}^*_p$ for which $$ s_1s_2 \equiv a \pmod{p} \quad \mbox{for some}\quad a \in [-X,X] \quad \mbox{for all}\quad s_1,s_2 \in A. $$ We also consider variations and generalizations.

math.NT↗
arXiv2019

The number of integer points close to a polynomial

Let $f(x)$ be a polynomial of degree $n \ge 1$ with real coefficients and let $X \ge 2$ and $δ\ge 0$ be real numbers. Let $\|\cdot\|$ be the distance to the nearest integer. We obtain upper bounds for the number of solutions to the inequality $\|f(x)\| \le δ$ with $x \in [X,2X] \cap \mathbb{N}$.

math.NT↗
arXiv2018

The larger sieve and polynomial congruences

We obtain a small improvement of Gallagher's larger sieve and we extend it to higher dimensions. We also obtain two interesting upper bounds for the number of solutions to polynomial congruences.

math.NT↗