al-Fārisī and the fundamental theorem of arithmetic

Abstract

The work of al-F~risi on amicable numbers begins with nine propositions of elementary number theory. The purpose of this article is to produce an English translation of these propositions and to provide a commentary on al-Ffirisi's methods. In particular we consider whether he proved, or attempted to prove the Fundamental Theorem of Arithmetic.

A Historical Survey of the Fundamental Theorem of Arithmetic

“…In [Agargün & Fletcher 1994] we produced an English translation of his first nine propositions and provided a commentary on al-Farisi's methods. The main aim of these nine propositions is to know and to find the divisors of a given number and hence is a preparation for the work on amicable numbers.…”

The fundamental theorem of arithmetic dissected

“…The FTA itself contains two results in one, the existence of a prime factorisation together with its uniqueness, and it is significant that Propositions VII.31 and VII.30 of the Elements lead immediately to their proofs although Euclid forbears to take these steps [1]. The first known proof of existence is due to al-Farisi (died c.1320), but he did not go on to prove uniqueness, mainly because his interest was in the divisors of a number rather than the factorisation itself [2,3]. And thereby hangs a tale.…”

"Al-Farisi and the Fundamental Theorem of Arithmetic, Ibn al-Hythem and Wilson Theorem (Historical Notes)"

“…Kamal al-Din al-Farisi, [6,7] work represents perhaps the most significant step toward the FTA made by mathematicians before Gauss [8] . His results appear in Tadhkirat al-Ahbab fıbayan al-tahabb (which means "memorandum for friends explaining the proof of amicability").…”