Wouter van Doorn

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math.NT2026

On the length of an interval that contains distinct multiples of the first $n$ positive integers

Wouter van Doorn

Confirming a conjecture by Erd\H os and Pomerance, we prove that there exist intervals of length $\frac{c n l o g n}{l o g l o g n}$ that do not contain distinct multiples of $1, 2, \ld…

math.NT2025

Growth rates of sequences governed by the squarefree properties of its translates

Wouter van Doorn, Terence Tao

We answer several questions of Erdős regarding sequences of natural numbers $A$ whose translates $n + A$ intersect with the squarefree numbers in various specified ways. For instance…

math.NT2025

Smooth sums with small spacings

Wouter van Doorn, Anneroos R. F. Everts

Solving a problem by Erdős, we prove that every positive integer $n$ can be written as a sum $n = b_{1} + b_{2} + \dots + b_{r}$ of distinct $3$ -smooth integers with $1 \le b_{1…

math.NT2025

Improved bounds for the Mayer-Erdős phenomenon on similarly ordered Farey fractions

Wouter van Doorn

Let $\frac{a _{1}}{b _{1}}, \frac{a _{2}}{b _{2}}, \dots$ be the Farey fractions of order $n$ . We then prove that the inequality $(a_{l} - a_{k}) (b_{l} - b_{k}) \geq 0$ holds for all $k$ and $l > k$ w…

math.NT2025

Partitions with prescribed sum of reciprocals: asymptotic bounds

Wouter van Doorn

In $1963$ Graham proved that every positive integer $n \geq 78$ can be written as a sum of distinct positive integers $a_{1}, a_{2}, \dots, a_{r}$ for which $\frac{1}{a_1} + \frac{1}{a_2…

math.NT2025

Partitions with prescribed sum of reciprocals: computational results

Wouter van Doorn

For a positive rational $α$ , call a set of distinct positive integers ${a_{1}, a_{2}, \dots, a_{r}}$ an $α$ -partition of $n$ , if the sum of the $a_{i}$ is equal to $n$ and the sum of th…

On the length of an interval that contains distinct multiples of the first n positive integers

Growth rates of sequences governed by the squarefree properties of its translates

Smooth sums with small spacings

Improved bounds for the Mayer-Erdős phenomenon on similarly ordered Farey fractions

Partitions with prescribed sum of reciprocals: asymptotic bounds

Partitions with prescribed sum of reciprocals: computational results

On the length of an interval that contains distinct multiples of the first $n$ positive integers