The author discusses the presentation of rational numbers as a sum of Egyptian fractions, i.e. fractions of the form , integers , and related problems. A number is called abundant, if the sum of all positive divisors of is . If is a vector of different primes and is a vector of nonnegative integers, then we write and the vector is called abundant, if some number of the form is abundant. The author shows that a necessary and sufficient condition for to be abundant is: .
He proves the following theorem. Suppose that is a fixed vector of successive primes with for some integer and is abundant. Suppose that for each integer with an equation of the form holds, where and are distinct. Then every rational positive number of the form has an Egyption fraction representation where distinct, of the form . As an example he shows and .