Wouter van Doorn

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20242026

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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 ErdÅs, we prove th…

math.NT2026

Gaps in Multiplicative Sidon Sets II

Wouter van Doorn, Quanyu Tang

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

math.NT2026

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.NT2026

Resolution of two conjectures by ErdÅs and Hall concerning separable numbers

Stijn Cambie, Wouter van Doorn

ErdÅs 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 di…

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.NT2026

Gaps in Multiplicative Sidon Sets

Wouter van Doorn, Pietro Monticone, Quanyu Tang

For a positive integer $n$ , let $g (n)$ denote the infimum of all real numbers $L$ such that there exists a multiplicative Sidon set $A \subseteq {1, 2, \dots, n}$ that intersects ever…

Consecutive integers free of certain prime factors

Gaps in Multiplicative Sidon Sets II

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

Resolution of two conjectures by ErdÅs and Hall concerning separable numbers

Three-term arithmetic progressions of consecutive powerful numbers

Gaps in Multiplicative Sidon Sets

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

Resolution of two conjectures by ErdÅs and Hall concerning separable numbers