On the Exponential Diophantine Equation 19p+ 31q = r2

Abstract

These days, Diophantine equations are used extensively in a wide range of mathematical fields, such as applied algebra, coordinate geometry, astronomy, and trigonometry. Diophantine equations are very difficult to solve, despite their significance. Not all Diophantine equations can be solved in the same way, unlike many other kinds of mathematical problems. Solving these equations is difficult and time-consuming since each one is handled independently. This study focuses on the Diophantine equation 19^P + 31^q = r²  where p, q, r are whole numbers. The purpose of this study is to determine whether this specific problem has a whole number solution.