Rational numbers whose sum and product are both integers

For n>۲, there are noninteger rational numbers r_i, \ i=۱, ۲, \ldots, n such that \sum_{i=۱} ^n r_i and \prod_{i=۱} ^n r_i are both integers. In this paper, while proving this proposition and with the help of the fundamental theorem of arithmetic and the properties of prime numbers, we will present an algorithm to generate numbers with this feature. Finally, we present a modified version of the original problem that requires further exploration.

کلیدواژه ها:

Blachket product ، Gauss's lemma ، Irreducible polynomials ، Primitive polynomials