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A congruence obstruction to Ibn al-Khawwām’s eleventh problem

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theorymath.HO — History and Overview

Contributed by Will Howe, Grant Huffman, Jensen Kohlmeyer ↗, Liam Kruer

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

We prove that there are no two rational squares whose sum is ten and such that adding its own square root to each gives another rational square. This is the existence question in Ibn al-Khawwām’s eleventh arithmetic problem. The proof uses only the residues of squares modulo three and a divisibility argument.

Provenance statement

OpenAI Codex developed the congruence argument, checked its algebra, and drafted the manuscript in response to the submitter’s instructions.

Tools used

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Codex

References

  1. K. I. A. Derouiche, A list of open problems in number theory posed by Ibn al-Khawwām (13th century): Historical and arithmetic analysis , preprint (2025), § 5.4, Problem 11. arXiv:2511.08400v1 (https://arxiv.org/abs/2511.08400v1).arXiv
  2. İ. Fazl oğlu, İbn el-Havvâm, eserleri ve el-Fevâid el-Bahâiyye fi el-Kavâid el-Hisâbiyye'deki çözümsüz problemler bahsi , in Osmanl Bilimi Araşt rmalar , F. Günergün (ed.), İstanbul, 1995, pp. 103--104, 123--124. Critical edition and commentary (https://isamveri.org/pdfdrg/D040138/1995/1995_FAZLIOGLUI.pdf).link
  3. R. Rashed, Histoire de l'analyse diophantienne classique: D'Abū Kāmil à Fermat , Scientia Graeco-Arabica, vol. 12, De Gruyter, Berlin--Boston, 2013, p. 143. doi:10.1515/9783110337884 (https://doi.org/10.1515/9783110337884).DOI

Version history

  1. v1 · 2 pagesSubmitted by Jensen KohlmeyerInitial depositCurrentNov 12, 2026