{
  "annotation_id": "43dde488-03d3-4d64-be4a-0027e537c782",
  "date_created": "2026-02-17T05:53:23.981000Z",
  "date_modified": "2026-02-17T05:53:23.981000Z",
  "file_hash": "86cdd17355a4462db9ac18b6e7dcfe2559e14089ed02272247df6cd71ce7b90e",
  "private": false,
  "record": {
    "abstract": "We develop a refinement process for two-term Machin-like formulas: $a_0 \\arctan{u_0} + a_1 \\arctan{u_1} = \\frac{\\pi}{4}$ (where $a_0 , a_1 \\in \\mathbb{Z}$, $u_0 , u_1 \\in \\mathbb{Q}_+^*$, $u_0 \u003e u_1$) by exploiting the continued fraction expansion of the ratio $\\alpha := \\frac{\\arctan{u_0}}{\\arctan{u_1}}$. This construction yields a sequence of derived two-term Machin-like formulas: $a_{- n} \\arctan{u_n} + a_{- n + 1} \\arctan{u_{n + 1}} = \\frac{\\pi}{4}$ ($n \\in \\mathbb{N}$) with positive rational arguments $u_n$ decreasing to zero and corresponding integer coefficients $a_{- n}$. We derive closed forms and estimates for $a_{-n}$ and $u_n$ in terms of the convergents of $\\alpha$ and prove that the associated rational sequence $(a_{- n} u_n + a_{- n + 1} u_{n + 1})_n$ converges to $\\pi/4$ with geometric decay. The method is illustrated using Euler\u0027s two-term Machin-like formula : $\\arctan(1/2) + \\arctan(1/3) = \\pi/4$.",
    "arxiv_id": "2601.10300",
    "authors": [
      "Bakir Farhi"
    ],
    "categories": [
      "math.NT"
    ],
    "license": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
    "title": "On refinements of two-term Machin-like formulas",
    "url": "https://arxiv.org/abs/2601.10300",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "41c97e3d-3371-435f-a207-f721a9b74327",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}