On some Diophantine Equations

We make use of the following two well-known lemmas :-

LEMMA (1 1). if (x, y) = 1, then x ± y and

 x^2r± x^2r-1y+x^2r-2 y²  +x^2r-3 y³+ …..y ^2r

cannot have any common factor except the di vision of r + 1.