The Solutions of the Diophantine Equations px

Let p and q be prime numbers. In this article, we show that all nonnegative integer solutions of the Diophantine equation p(x)+p(y) = z(q) are (p, q, x, y, z) = (2, q, qt+q-1, qt+q-1, 2(t+1)), (2(q) -1, q, qt+ 1, qt, 2(2(q) - negative integer solutions of the Diophantine equation p(x) - p(y) = z(q) are (p, q, x, y, z) = (p, q, t, t, 0), (2, q, qt + 1, qt, 2(t)), (4v(2) + 1, 2, 2t + 1, 2t, 2v(4v(2) + 1)(t)), (3,3,3t + 2,3t, 2 center dot 3(t)), where t is a non-negative integer and v is a positive integer.