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

TL;DR

This paper compiles and analyzes thirty-three open problems in 13th-century number theory posed by Ibn al-Khawwam, providing historical context, arithmetic insights, and some original solutions, thus enriching the understanding of the development of number theory.

Contribution

It presents a comprehensive list of historical number theory problems from Ibn al-Khawwam, with analysis, solutions, and contextualization within mathematical history.

Findings

Historical problems in number theory identified and analyzed.

Some problems have been solved or partially solved.

The work connects 13th-century problems to modern number theory developments.

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.