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This book is aimed at students, researchers and anyone with an interest in mathematics. Some aptitude is needed for general readers. It will be particluarly useful to students who wish to do research in patterns of numbers and solving polynomial equations for integer solutions. In this book,different approaches of solving quadratic equations with four unknowns have been studied. The above methods may also be applied to multidegree multivariate polynomial diophantine equations with suitable modifications. A number of applications involving numerical relations for second order Ramanujam numbers have been ilustrated in this book which will motivate researchers seeking for problems in the field of diophantine equations and number patterns.