Analysis of Generalization of a Problem of Vieta’s Descend Method with Examples and Computing Support

Abstract

Let  and  be fixed integer numbers. Assume that (a2+b2+c)  is divisible by (ab+d) for some natural numbers a and b. Then the value of the fraction $$k ( = {(a^2+b^2+c) \over (ab+d)})$$ remains the same. Statement of this kind will be proved in pp. 1-3 and illustrated on some examples in pp. 3-10. The general method of proofs will be unified and simplified. Computing support will be provided: in pages 11-19 a simple program code is defined with the help of which one can hunt for natural numbers a, b with the same integer values of c,  d and k. Here, a number of examples are given as well.