Product representation of perfect cubes
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866911882250551296 |
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| author | Fleiner, Zsigmond György Juhász, Márk Hunor Kövér, Blanka Pach, Péter Pál Sándor, Csaba |
| author_facet | Fleiner, Zsigmond György Juhász, Márk Hunor Kövér, Blanka Pach, Péter Pál Sándor, Csaba |
| contents | Let $F_{k,d}(n)$ be the maximal size of a set ${A}\subseteq [n]$ such that the equation \[a_1a_2\dots a_k=x^d, \; a_1<a_2<\ldots<a_k\] 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 years old conjecture of Verstraëte.
We 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_12088 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Product representation of perfect cubes Fleiner, Zsigmond György Juhász, Márk Hunor Kövér, Blanka Pach, Péter Pál Sándor, Csaba Combinatorics Number Theory Let $F_{k,d}(n)$ be the maximal size of a set ${A}\subseteq [n]$ such that the equation \[a_1a_2\dots a_k=x^d, \; a_1<a_2<\ldots<a_k\] 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 years old conjecture of Verstraëte. We 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. |
| title | Product representation of perfect cubes |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2405.12088 |