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Claim. For an interval $[u,v]$ of positive integers write $H([u,v])=\sum_{n=u}^v1/n$. Theorem 3.2 of the note states that if two intervals $I,J$ intersect and $H(I)+H(J)$ is an integer, then, up to swapping them, the pair is one of $([1,1],[1,1])$, $([1,2],[1,2])$, $([1,2],[2,2])$ and $([2,2],[2,2])$, and that all four pairs have an integer sum. So, in the formulation of Problem 288, which lets the two intervals overlap or coincide, the pairs with intersecting intervals are finitely many. The note's argument: the union of two intersecting intervals is an interval $[r,s]$ on which the sum has coefficients $1$ or $2$; when $s\ge3$ and the union has at least two elements, Bertrand's postulate or the Sylvester--Schur theorem gives an odd prime dividing exactly one denominator in $[r,s]$ (its Lemma 3.1), whose term is then the unique term of lowest $p$-adic valuation, so the sum is not an integer; when the union is a single element $\{r\}$, both intervals are $[r,r]$ and the sum $2/r$ is an integer only for $r=1,2$; the remaining unions inside $\{1,2\}$ are checked directly. The note's Sections 4 to 18 treat the disjoint case, with intervals $[a,b]$ and $[c,d]$, $b<c$: every disjoint solution has upper length $d-c+1<b$, in fact $o(b)$ along any infinite family, and every denominator $c,\ldots,d$ divides $\operatorname{lcm}(1,\ldots,b)$; mirror congruences for large prime powers and a lower bound for the denominator of $H([c,d])$ follow, and for fixed upper length $L=2$ (its Section 14) a solution must have $c=P_0u_0$, $c+1=P_1u_1$ with $P_0,P_1$ carrying almost all of the large primes and $u_0,u_1$ small smooth cofactors. The note states that it does not claim the disjoint case.

Submission note. Posted to the site's forum by Ritvik Nayak on 3 May 2026:

I've put together a brief research note on this problem, with a great deal of assistance from GPT 5.5 Thinking.

The research note is only partial progress toward a complete solution, but it solves the intersecting case in its entirety, and then gives several obstructions in the disjoint case using $p$-adic methods, denominator inequalities, and CRT/smoothness conditions.

For example, for any disjoint solution with intervals $[a,b]$ and $[c,d]$, the upper denominators must divide

$$> \Lambda_b = \operatorname{lcm}(1,2,\ldots,b). >$$

In the sufficiently large fixed length case, the remaining problem is reduced to a pretty rigid smooth CRT condition. In the special case $L=2$, this gives an obstruction of the form

$$> c = P_0u_0,\qquad c+1 = P_1u_1,\qquad P_1u_1-P_0u_0=1, >$$

where the $P_i$ contain most of the large prime factors, while the $u_i$ are small smooth cofactors.

I would really appreciate any comments, especially on whether the $L=2$ obstruction can be pushed further.

Covers. The pairs of intervals that intersect: among them exactly four, all inside $\{1,2\}$, have an integer sum. Not covered: the disjoint pairs, for which the note proves necessary conditions and no finiteness; the problem's statement, and its singleton case with the singleton outside the other interval, remain open.

Standing. Claimed. The note, "A Research Note on Harmonic Sums over Two Integer Intervals: Intersecting intervals and reductions in the disjoint case", dated May 2026 and written, by its title page, with assistance from GPT-5.5 Thinking, was announced in the problem's discussion thread on 3 May 2026 by its author, who is the claimant; it is a PDF on Google Drive with no arXiv or journal record. It is not on the site's proof-claim tab, which is empty, and the site's label is OPEN, so no curator, referee or named mathematician has accepted it. Two replies of the same day suggest a compression of the length-$2$ condition and report that an automated check flagged one minor issue in the note, without saying which. The statement of Theorem 3.2 and the structure of its proof are compiled above; the corpus records no check of the proof.