On the Extraction of Amicable Pairs Between Ibn Sina-al-Baghdadi and al-Kashi

TL;DR

This paper analyzes medieval Arabic mathematicians Ibn Sina, al-Baghdadi, and al-Kashi's methods for extracting amicable pairs, revealing differences from Thabit ibn Qurra's rule and identifying unresolved conjectures and errors in their statements.

Contribution

It clarifies the distinct approaches of Ibn Sina, al-Baghdadi, and al-Kashi to amicable pairs, contrasting them with Thabit ibn Qurra's rule, and highlights their implications for unresolved conjectures and errors.

Findings

Ibn Sina and al-Baghdadi's statements lead to an unsolved conjecture.

Al-Kashi's statement results in an incorrect amicable pair (2024, 2296).

No counterexamples found among known 51 Mersenne primes.

Abstract

In this note, we briefly analyse the works on the extractions of amicable pairs in some medieval Arabic literature. In this note, we briefly analyse the works Ibn Sina (c.980-1037), al-Baghdadi (c.980-1037) and al-Kashi (d. c. 1429) on the extraction of amicable pairs. We compare these works in the view of the well-known Thabit ibn Qurra's rule. Contrary to the belief that these authors are merely stating variations of the famous Thabit ibn Qurra's (d. 901) rule in different manners, we show that their statements are different. We prove that the statements of both ibn Sina and al-Baghdadi led to an unsolved conjecture, while al-Kashi's statement led to the wrong amicable pair (2024, 2296). The conjecture implied by ibn Sina and al-Baghdadi statements requires testing big primes corresponding to the known 51 Mersenne primes, and we show that no counterexamples yet exist.