Paper Title
A COMPREHENSIVE OVERVIEW OF DIOPHANTINE EQUATIONS, INCLUDING EARLY SOLUTIONS AND FAMOUS CONJECTURES, TRACING THEIR ORIGINSAbstract
Solving the Diophantine equation has fascinated mathematicians from various civilizations. In this paper, we propose the resolution of quadratic Diophantine equations with integer coefficients. Our contribution consists of generalizing certain results which have already been developed in the literature. This paper also proposes the criterion on the solvability of the quadratic Diophantine equation here studied. We consider the Diophantine equation of the title which was re- cently solved, in terms of the number of solutions to it for a 2 {1,2,4}, in (1). However, a counterexample was provided in (6). We provide another coun- terexample and show that both the example herein and the one in (6) are results of Ljunggren (7) from the early 1940s. Given that these are all the omissions from (1), this secures the study of the equation in the title. For relatively prime D1,D2 2 N (the natural numbers), D = D1D2, k 2 N, prime to D, ‚ 2 {1, p 2,2}, with ‚ = 2 if k is even, the Diophantine equation (1) D1x2 + D2 = ‚2kn; x,n 2N
KEYWORDS : Biquadrates; Quartic diophantine equation, Homogeneous Ternary Quadratic, Integral solutions