Two problems of number theory in Islamic times (Q1205951)

The study under review discusses briefly two aspects of the number theory in Islamic times. Thus, as early as the ancient Greek, mathematicians tried to determine the perfect number, i.e. a number \(n\) so that \(s(n)=n\). Later on, based on Euclid's formula, Euler demonstrated that any even perfect number must have the form proposed (if \(2^ m\) is a prime, then \(2^{m-1}(2^ m-1)\) is perfect), which means that Euclid's demonstration depends on the existence of odd perfect numbers. Also, the Pythagoreans dealt with the so-called amicable numbers, pairs of numbers in which each is the sum of the aliquot parts of the other. In Islamic times there existed a simple rule for the discovery of the so-called balanced numbers. It was first mentioned in the study of Abu-Manşur Al-Baghdadi, entitled ``Completion of arithmetic'', as well as, later on, in the first half of the seventeenth century, at the Persian mathematician Muhammad Baqir Yazdi. Very surprising is the fact that simple rules and calculations still raise unsolved questions, even today.