Integral Solution to the Equation x2+2a.7b=yn - Yow, Kai Siong (2011)
Diophantine equation is an equation in which solutions to it are from some predetermined classes and it is one of the oldest branches of number theory. There are many types of diophantine equations, for instance linear diophantine equation, exponential diophantine equation and others. In this research, we will investigate and find the integral solutions to the diophantine equation x² +2a.7b=yⁿ where a and b are positive integers and n is even. By fixing n = 2r , we determine the generators of and x and yr for 1 ≤ a ≤ 6 with any values of b. Then, we investigate the necessary conditions to obtain integral solutions of x and y under each value of af there is any. The approach is by looking at the possible combinations for the product 2a⋅7b and solving the equations simultaneously. Then, from the results obtained, we substitute the values of a followed by b to get integer values of and a and yr under each category. After that, the equations are grouped according to the pattern that emerged and a geometric progression formula is applied to create the general formulae for the generators of solutions to the equation examined. Besides that, we have to identify the range of i, the number of non-negative integral solutions associated with each b for different values of a. When b is even, we find some special cases of determining the generators of solutions for and x and yr with a certain condition. From our investigation, we find that there is no integral solution of and x and yr the diophantine equation x² +2a.7b=yⁿ, when n is even and a = 1. It is found that the number of generators to determine the integral solutions to the equation depend on the values of a. Values of y are determined by taking the r-th root of yr for certain values of r.