On the Extraction of Amicable Pairs Between Ibn Sīnā- al-Baghdādī and al-Kāshī

Abstract In this note, we briefly analyse the works Ibn Sīnā (c.980-1037), al-Baghdādī (c.980-1037) and al-Kāshī (d. c. 1429) on the extraction of amicable pairs. We compare these works in the view of the well-known Thābit ibn Qurrā’s rule. Contrary to the belief that these authors are merely stating variations of the famous Thābit ibn Qurrā’s (d. 901) rule in different manners, we show that their statements are different. We prove that the statements of both ibn Sīnā and al-Baghdādī led to an unsolved conjecture, while al-Kāshī’s statement led to the wrong amicable pair . The conjecture implied by ibn Sīnā and al-Baghdādī statements requires testing big primes corresponding to the known 51 Mersenne primes, and we show that no counterexamples yet exist.