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DTSTART;TZID=Asia/Seoul:20251125T163000
DTEND;TZID=Asia/Seoul:20251125T173000
DTSTAMP:20260416T124008
CREATED:20250929T123857Z
LAST-MODIFIED:20251104T141401Z
UID:11664-1764088200-1764091800@dimag.ibs.re.kr
SUMMARY:Péter Pál Pach\, Product representation of perfect cubes
DESCRIPTION:Let $F_{k\,d}(n)$ be the maximal size of a set ${A}\subseteq [n]$ such that the equation \[a_1a_2\cdots a_k=x^d\, \; a_1<a_2<\ldots<a_k\] has no solution with $a_1\,a_2\,\ldots\,a_k\in A$ and integer $x$. Erdős\, Sárközy and T. Sós studied $F_{k\,2}$\, and gave bounds when $k=2\,3\,4\,6$ and also in the general case. We study the problem for $d=3$\, and provide bounds for $k=2\,3\,4\,6$ and $9$\, furthermore\, in the general case\, as well. In particular\, we refute an 18-year-old conjecture of Verstraëte. \nWe also introduce another function $f_{k\,d}$ closely related to $F_{k\,d}$: While the original problem requires $a_1\, \ldots \, a_k$ to all be distinct\, we can relax this and only require that the multiset of the $a_i$’s cannot be partitioned into $d$-tuples where each $d$-tuple consists of $d$ copies of the same number. \nJoint work with Fleiner\, Juhász\, Kövér and Sándor.
URL:https://dimag.ibs.re.kr/event/2025-11-25/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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