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 x/y is asymptotically equal to (3 + √5)/2 or (3 − √5)/2. Results of the paper are based on data obtained via Computer Algebra System ( 5). Some procedures presented in the paper made it possible to discover interesting regularities concerning simple continued fractions of certain numbers.