Erdős Problem #317: independent continuation
Date: 23 September 2026. Source versions read in full:
317.tex, 317_v2.tex in
gpt_pro_5.4.
1) OBJECTIVE AND ALL-VERSION AUDIT
Both supplied versions contain the same meet-in-the-middle argument and numerical table. Describing an algorithm does not independently certify its reported output. Here I give short modular certificates for the strict lower-bound question that do not require enumerating $3^{n/2}$ signed sums.
2) A PRIME CERTIFICATE EXCLUDING NUMERATOR ONE
Let $L=\operatorname{lcm}(1,\ldots,n)$ and let $p$ be a prime with $n/2<p\le n$. Only the denominator $p$ is divisible by $p$. Reducing a scaled signed sum modulo $p$ yields \[
L\sum_{k=1}^n\frac{\delta_k}{k}\equiv\delta_p L/p\pmod p,
\qquad\delta_k\in\{-1,0,1\}.
\] Therefore a scaled numerator equal to $1$ or $-1$ is impossible whenever \[
L/p\not\equiv1,-1\pmod p. \tag{1}
\] Since every nonzero scaled numerator is an integer, any prime certificate $1$ proves the strict bound $|\sum\delta_k/k|>1/L$.
3) EXACT CERTIFICATES FOR THE SUPPLIED NEW RANGE
The following small residues, recomputed with integer arithmetic, certify all six of the source's additional cases: \[
\begin{array}{c|rrrrrr}
n&27&28&29&30&31&32\\\hline
p&19&19&17&17&17&17\\
(L/p)\bmod p&4&4&5&5&2&4
\end{array}
\] Each residue is different from $1$ and $p-1$. These certificates establish only that the minima exceed $1/L$; they do not independently certify the much larger exact minima in the source's table. More generally, reduction modulo any prime $p\le n$ isolates the denominators whose $p$-adic valuation equals that of $L$. The possible residues of their signed contributions form a finite set; excluding $1$ and $-1$ from that set gives the same conclusion, even if several denominators survive.
4) REMAINING GAP
I have not proved that a certifying prime or a stronger modular obstruction exists for every large $n$. This route concerns the strict lower bound only; it supplies no $O(2^{-n})$ upper bound for the minimum nonzero signed sum. The cardinality argument in the sources already appears in the official discussion and is not a new general estimate here.
7) FINAL
UNRESOLVED. Several large exhaustive-search claims are replaced, for one of their conclusions, by tiny independently reproducible certificates. Discussion checked:
https://www.erdosproblems.com/forum/thread/317. Verification:
verification/root_checks.py.
8) COMPLETION ESTIMATE
COMPLETION: 10% This is a subjective estimate of progress toward the entire stated problem, not a probability that the conjecture or an unverified proof is correct.