Erdős Problem #290 — Decreasing Denominators of Harmonic Block Sums

The problem asks whether, for every starting point $a$, some longer consecutive reciprocal sum has a smaller denominator in lowest terms. Paul Erdős and Ronald Graham posed the question; Peter Shiu and Wouter van Doorn gave independent affirmative answers.

Known results

• For $a=1$, denominator drops occur infinitely often; in particular, $d_n<d_{n-1}$ when $n=p(p-1)$ for the relevant primes $p$.
• For general $a$, the least drop point $b(a)$ satisfies $b(a)\le 4.374(a-1)$ for all $a\ge 6$.

Van Doorn’s 2024 resolution

Van Doorn proved that every $a\in\mathbb{N}$ has infinitely many $b>a$ for which the reduced denominator of $\sum_{i=a}^{b}1/i$ is smaller than that of $\sum_{i=a}^{b-1}1/i$. He also obtained $0.54<\liminf_{a\to\infty}(b(a)-a)/\log a<0.61$.

Current status (as of June 2026): The ordinary harmonic-interval problem is resolved affirmatively, with infinitely many drops for every $a$; quantitative bounds are known for the first drop.