A List of Open Problems in Number Theory Posed by Ibn al-Khaww\=am (13th Century): Historical and Arithmetic Analysis

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-Khaww\=am al-Baghd\=ad\=i (Abdall\=ah ibn Muhammad ibn Muhammad al-Khaww\=am), extracted from his arithmetic treatise al-Faw\=aid al-Bah\=aiyya f\=i qaw\=aid al-his\=abiyya. 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.