Infinitely many positive solutions of the diophantine equation x2 − kxy + y2 + x = 0

We prove that the equation x2 − kxy + y2 + x = 0 with k ϵ N+ has an infinite number of positive integer solutions x and y if and only if k = 3. For k = 3 the quotient xy is asymptotically equal to (3 + √5)/2 or (3 − √5)/2. Results of the paper are based on data obtained via Computer Algebra System (derive 5). Some derive procedures presented in the paper made it possible to discover interesting regularities concerning simple continued fractions of certain numbers.