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This book is aimed at students, researches and anyone with an interest in numbers and equations.Some aptitude is needed for general readers. It will be particularly useful to students who wish to do research in number patterns and solving polynomial equations for integer(Real or Gaussian) solutions. In this book, Real as well as Gaussian integer solutions for the Pythagorean equation are presented. The above methods may also be apply to multidegree multivariate polynomial diophantine equations with suitable modifications. A number of applications involving pythagorean equation have been illustrated in this book which will motivate students and researchers seeking for problems in the field of Quadratic Diophantine equations and Number patterns.