On Some Diophantine Equations (i)

In this paper we study the equation m−n = py,where p is a prime natural number, p≥ 3. Using the above result, we study the equations x + 6pxy + py = z and the equations ck(x 4 + 6pxy + py) + 4pdk(x y + pxy) = z, where the prime number p ∈ {3, 7, 11, 19} and (ck, dk) is a solution of the Pell equation, either of the form c −pd = 1 or of the form c − pd = −1. I. Preliminaries. We recall some necessary results. Proposition 1.1. ([3],p.74)The integer solutions of the Diophantine equation x1 + x 2 2 + ... + x 2 k = x 2 k+1 are the following ones    x1 = ±(m1 + m2 + ... + mk−1 −mk) x2 = 2m1mk ..................... ..................... xk = 2mk−1mk xk+1 = ±(m1 + m2 + ... + mk−1 + mk), with m1, ..., mk integer numbers. From the geometrical point of view, the solutions (x1, x2, ..., xk, xk+1) are the sizes x1, x2, ..., xk of a right hyper-parallelipiped in the space R and xk+1 is the length of its diagonal. Proposition 1.2. ([1], p.150) The quadratic field Q( √ d), where d ∈ N∗ , d is square free, has an Euclidean ring of integers A (with respect to the norm N), for d ∈ {2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73} .