Numerous researches have been devoted in finding the solutions (, ,), in the set of non-negative integers, of Diophantine equations of type + = 2 (1), where the values and are fixed. In this paper, we also deal with a more generalized form, that is, equations of type + = 2 (2), where is a positive integer. We will present results that will guarantee the non-existence of solutions of such Diophantine equations in the set of positive integers. We will use the concepts of the Legendre symbol and Jacobi symbol, which were also used in the study of other types of Diophantine equations. Here, we assume that one of the exponents is odd. With these results, the problem of solving Diophantine equations of this type will become relatively easier as compared to the previous works of several authors. Moreover, we can extend the results by considering the Diophantine equations + 1 1 2 2 … = 2 (3) in the set of positive integers.
In this paper, we solve the Diophantine equation px + (p + 4k)y = z2 in N0 for prime pairs (p, p+ 4k). First, we consider cousin primes p and p+ 4. Then we extend the study to solving px + (p + 4)y = z 2n, where n ∈ N\{1}. Furthermore, we solve the equation px + (p + 4k)y = z2 for k ≥ 2. As a result, we show that this equation has a unique solution (p, p + 4k, x, y, z) =(3, 11, 5, 2, 122) whenever x 1 and y 1. Finally, we show the finiteness of number of solutions in N.
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