W. V. van Doorn

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20242026

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math.NT2026

The cardinality of a set containing the pairwise sums of four positive integers

Wouter van Doorn, John Erlbacher

Choi, Erdős and Szemerédi showed that there exists an absolute constant $C$ such that for all subsets $A \subseteq {1, 2, \dots, 2 n}$ with at least $n + C$ elements, there exist f…

math.NT2026

The shortest harmonic sums with decreasing denominator

Wouter van Doorn

For a positive integer $a$ , let $b (a)$ be the smallest integer $b > a$ such that the denominator of $\frac{1}{a} + \frac{1}{a + 1} + \dots + \frac{1}{b}$ is smaller than the denomin…

math.NT2026

Consecutive integers free of certain prime factors

Wouter van Doorn, Quanyu Tang

Let $n_{k}$ denote the least integer $n > 2 k$ such that $(n - k) (n - k + 1) \dots (n - 1)$ is not divisible by any prime in the interval $(k, 2 k)$ . Confirming a conjecture of Erdős, we prove tha…

math.NT2026

Gaps in Multiplicative Sidon Sets II

Wouter van Doorn, Quanyu Tang

With $ρ = \frac{13 - 69}{10} \approx 0.47$ , it was recently established that there exist multiplicative Sidon sets (sets without any non-trivial solutions to $ab = c d$ ) in $\{1…

math.NT2026

Three-term arithmetic progressions of consecutive powerful numbers

Wouter van Doorn

We show that infinitely many three-term arithmetic progressions $N, N + d, N + 2 d$ of powerful numbers exist with $d = 2 N + 1$ . We further conjecture that infinitely many of the…

math.NT2025

The smallest denominator not contained in a unit fraction decomposition of $1$ with fixed length

Wouter van Doorn, Quanyu Tang

Let $v (k)$ be the smallest integer larger than $1$ that does not occur among the denominators in any identity of the form $1 = \frac{1}{n _{1}} + \dots + \frac{1}{n _{k}},$ where $1 \le n_1<\…

math.NT2025

Resolution of two conjectures by Erdős and Hall concerning separable numbers

Stijn Cambie, Wouter van Doorn

Erdős and Hall defined a pair $(m, n)$ of positive integers to be interlocking, if between any pair of consecutive divisors (both larger than $1$ ) of $n$ (resp. $m$ ) there is a div…

math.NT2025

Lacunary sequences whose reciprocal sums represent all rational numbers in an interval

Wouter van Doorn, Vjekoslav Kovač

Disproving a conjecture of Bleicher and Erdős, we show that there exists a lacunary sequence of positive integers such that finite sums of reciprocals of its terms attain all ratio…

math.NT2024

On the congruence properties and growth rate of a recursively defined sequence

Wouter van Doorn

Let $a_{1} = 1$ and, for $n > 1$ , $a_{n} = a_{n - 1} + a_{⌊ \frac{n}{2} ⌋}$ . In this paper we will look at congruence properties and the growth rate of this sequ…

The cardinality of a set containing the pairwise sums of four positive integers

The shortest harmonic sums with decreasing denominator

Consecutive integers free of certain prime factors

Gaps in Multiplicative Sidon Sets II

Three-term arithmetic progressions of consecutive powerful numbers

The smallest denominator not contained in a unit fraction decomposition of 1 with fixed length

Resolution of two conjectures by Erdős and Hall concerning separable numbers

Lacunary sequences whose reciprocal sums represent all rational numbers in an interval

On the congruence properties and growth rate of a recursively defined sequence

The smallest denominator not contained in a unit fraction decomposition of $1$ with fixed length