On product representations of squares
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866914986446553088 |
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| author | Tao, Terence |
| author_facet | Tao, Terence |
| contents | Fix $k \geq 2$. For any $N \geq 1$, let $F_k(N)$ denote the cardinality of the largest subset of $\{1,\dots,N\}$ that does not contain $k$ distinct elements whose product is a square. Erdős, Sárkőzy, and Sós showed that $F_2(N) = (\frac{6}{π^2}+o(1)) N$, $F_3(N) = (1-o(1))N$, $F_k(N) \asymp N/\log N$ for even $k \geq 4$, and $F_k(N) \asymp N$ for odd $k \geq 5$. Erdős then asked whether $F_k(N) = (1-o(1)) N$ for odd $k \geq 5$. Using a probabilistic argument, we answer this question in the negative. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_11610 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On product representations of squares Tao, Terence Number Theory 11N25 Fix $k \geq 2$. For any $N \geq 1$, let $F_k(N)$ denote the cardinality of the largest subset of $\{1,\dots,N\}$ that does not contain $k$ distinct elements whose product is a square. Erdős, Sárkőzy, and Sós showed that $F_2(N) = (\frac{6}{π^2}+o(1)) N$, $F_3(N) = (1-o(1))N$, $F_k(N) \asymp N/\log N$ for even $k \geq 4$, and $F_k(N) \asymp N$ for odd $k \geq 5$. Erdős then asked whether $F_k(N) = (1-o(1)) N$ for odd $k \geq 5$. Using a probabilistic argument, we answer this question in the negative. |
| title | On product representations of squares |
| topic | Number Theory 11N25 |
| url | https://arxiv.org/abs/2405.11610 |