Konstantine Zelator. The Rational Number n/p as a sum of two unit fractions

Natural Sciences / Mathematics / Algebra

Submitted on: Apr 12, 2012, 05:23:24

Description: In a 2011 paper published in the journal "Asian Journal of Algebra"(see reference[1]), the authors consider, among other equations,the diophantine equations 2xy=n(x+y) and 3xy=n(x+y). For the first equation, with n being an odd positive integer, they give the solution x=(n+1)/2, y=n(n+1)/2. For the second equation they present the particular solution, x=(n+1)/3,y=n(n+1)/3, where is n is a positive integer congruent to 2modulo3. If in the above equations we assume n to be prime, then these two equations become special cases of the diophantine equation, nxy=p(x+y) (1), with p being a prime and n a positive integer greater than or equal to 2. This 2-variable symmetric diophantine equation is the subject matter of this article; with the added condition that the intager n is not divisible by the prime p. Observe that this equation can be written in fraction form: n/p= 1/x + 1/y