A List of Open Problems in Number Theory Posed by Ibn al-Khawwām (13th Century): Historical and Arithmetic Analysis
K. I. A Derouiche
TL;DR
This work collects and analyzes Ibn al-Khawwām's 33 arithmetic open problems from al-Fawā'id al-Bahā'iya, pairing historical context with detailed arithmetic examination and occasional original solutions. It reproduces the problem list and supplements it with bibliographic notes, situating the collection within the broader history of number theory and Islamic mathematics. The study classifies the problems into congruence, higher‑degree polynomial equations, Diophantine problems, and multiplicative questions, illustrating ancient methods (e.g., variable changes, Cardano–Ferrari techniques) and pointing to connections with later developments such as Galois theory. By tracing commentators and subsequent scholarship (notably Abdeljouad and Hedfi, among others), the paper underscores the enduring significance of these medieval questions for both math history and contemporary number theory research.
Abstract
Mathematicians have long been fascinated by the resolution of algebraic and Diophantine equations in search of integer or rational solutions. This article presents a list of thirty-three open problems in number theory, posed in the 13th century by Ibn al-Khawwām al-Baghdādī (Abdallāh ibn Muhammad ibn Muhammad al-Khawwām), extracted from his arithmetic treatise al-Fawāid al-Bahāiyya fī qawāid al-hisābiyya. We provide a historical and arithmetic analysis of these problems and their solutions, and, whenever possible, offer original solutions and bibliographic notes. This work situates these problems within the broader development of number theory.