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This book is to identify possible solutions of the Diophantine equation by there are research five papers in this book, the first three papers are the development to find solutions of the Diophantine equation x^2 + k y^2 = z^2 for k = 7, k is a prime number and k is a integer number, and the last two papers are shown that the Diophantine equation 2^x + 47^y = z^2 has only two solutions (x,y,z) in non-negative integers and the Diophantine equation 13^x + 10^y + 2016^z = w^2 has no non-negative integer solution.