A Collection of Special Binary and Ternary Quadratic Diophantine Equation with Integer Solutions and Properties

One of the areas of Number theory that has attracted many mathematicians since antiquity is the subject of diophantine equations. A diophantine equation is a polynomial equation in two or more unknowns such that only the integer solutions are determined. No doubt that diophantine equation possess supreme beauty and it is the most powerful creation of the human spirit. A pell equation is a type of non-linear diophantine equation in the form where  and square-free. The above equation is also called the Pell-Fermat equation. In Cartesian co-ordinates, this equation has the form of a hyperbola. The binary quadratic diophantine equation having the form

Eqn.

is referred to as the positive form of the pell equation and the form

Eqn. 

is called the negative form of the pell equation or related pell equation. It is worth to remind that (2) is solvable for only certain values of D and always in the case of (1). An obvious generalisation to the Pell equation is the equation of the form  which is known as Pell-like equation.

 Pell equations arise in the investigation of numbers which are figurate in more than one way, for example, simultaneously square & triangular and as such they are extremely important in Number theory. In the solution of cubic equation and in certain other situations it is desirable to have a method for extracting the cube root of a binomial surd. This may be accomplished by the aid of the pell equation. We use pell equation to solve Archimedes’ Cattle problem. Pell’s equation is connected to algebraic number theory, Chebyshev polynomials and continued fractions. Other applications include solving problems involving double equations, rational approximations to square roots, sums of consecutive integers, Pythagorean triangles with consecutive legs, consecutive Heronian triangles, sums of  and  consecutive squares and so on. Man’s love for numbers is perhaps older than number theory. The love for large numbers may be a motivation for pellian equation.

 In studies on Diophantine equations of degree two with two and three unknowns, significant success was attained only in the twentieth century. There has been interest in determining all solutions in integers to quadratic Diophantine equations among mathema6ticians.

The main thrust in this book is on solving second degree Diophantine equations with two and three variables. This book contains a reasonable collection of special quadratic Diophantine problems in two and three variables distributed in 12 chapters. The process of getting different sets of integer solutions to each of the quadratic Diophantine equations considered in this book are illustrated in an elegant manner. The articles with solutions and properties presented in chapters 1, 2 & 3 are Pell equations and in chapters 4,5 &6 are Pell-like equations. The articles with solutions presented in chapters 7-12 are quadratic equations with three unknowns of the form . In Cartesian co-ordinates, this equation has the form of a right circular cone.