An algorithm for solving a certain class of Diophantine equations. I

Abstract

Abstract. A class of Diophantine equations is defined and an algorithm for solving each equation in this class is developed. The methods consist of techniques for the computation of an upper bound for the absolute value of each solution. The computability of these bounds is guaranteed. Typically, these bounds are well within the range of computer programming and so they constitute a practical method for computing all solutions to the Diophantine equation in question. As a first application, a bound for a cubic equation is computed. As a second application, a set of quartic equations is studied. Methods are developed for deriving various sets of conditions on the coefficients in such equations under which a bound exists and can be computed.

An Algorithm for Solving a Certain Class of Diophantine Equations. I

“…In Part II of this paper (Hilliker [3]) we shall turn to more realistic examples. There, as a further illustration of the techniques, we shall study the general quartic polynomial Diophantine equation with integer coefficients, in two variables.…”

“…For more on Runge's Theorem, see, in addition to Hilliker [3], and Runge [ 13], the works of Hilliker and Straus [5], [6], Maillet [8], [9], Mordell [11], Schinzel [14], and Skolem [15], [16].…”

“…An example of this would be 1/x + \/y = k. For k = 2, this equation has only the solution (1,1); but if k varies over the interval [1,2], in the infinite sequence k = 2, 3/2, 4/3, 5/4,..., there are infinitely many solutions (1,1), (1,2), (1,3), (1,4),... . Our objective will be to characterize the classes A and A, in terms of the existence of a certain quantity, q = q(xx, x2,...,x"), defined for all (x,, x2,...,x") in D. Then, based upon this characterization, we shall define another class A2, a subclass of A,, which will turn out to contain only Diophantine equations that possess a computable bound.…”

“…In order to simplify the proof we shall reduce the problem of solving the equation (1) It is then realized that our original Diophantine equation (1) with k i= 0 is equivalent to (3) xy2 + ay + k = 2x3…”

“…The Case k < 0. Let us now assume that k < 0 in the equation (3). Then y is given by (4), since the inequality (5) now automatically holds.…”