{
  "annotation_id": "1c635d46-db64-461f-9d3d-d2e305b5045b",
  "date_created": "2026-02-17T05:53:08.811000Z",
  "date_modified": "2026-02-17T05:53:08.811000Z",
  "file_hash": "c63d0a78dc2a18fb0b1fd59ebc8f8926fa8d9594e67594b0ce607166f8db935d",
  "private": false,
  "record": {
    "abstract": "Let $\\rho_k(N)$ denote the maximum size of a set $A\\subseteq \\{1,2,\\dots,N\\}$ such that no product of $k$ distinct elements of $A$ is a perfect $d$-th power. In this short note, we prove that $\\rho _d(N)=\\sum\\limits_{k=1}^{d-1}\\pi\\left( \\frac{N}{k} \\right) +O_d(\\pi (N^{1/2}))$, furthermore, for prime power $d$ and sufficiently large $N$ we have $\\rho _d(N)=\\sum\\limits_{k=1}^{d-1}\\pi\\left( \\frac{N}{k} \\right)$. This answers a question of Verstra\\\"ete.",
    "arxiv_id": "2601.07000",
    "authors": [
      "P\u00e9ter P\u00e1l Pach",
      "Csaba S\u00e1ndor"
    ],
    "categories": [
      "math.CO",
      "math.NT"
    ],
    "license": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
    "title": "Product representations of perfect powers",
    "url": "https://arxiv.org/abs/2601.07000",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "cc13cf9c-1040-4e83-b26e-cc864c454985",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}