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

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.