  In this paper, we solve the simultaneous Diophantine equations(SDE)
x_1^u+...+x_n^u=k(y_1^u+...+y_{n/k}); u=1,3, where n >3, and k< n, is a divisor
of n , and obtain nontrivial parametric solution for them. Furthermore we
present a method for producing another solution for the above Diophantine
equation (DE) for the case u = 3, when a solution is given. We work out some
examples and find nontrivial parametric solutions for each case in nonzero
integers. Also we prove that the other DE
p_1x_1^{a_1}+....+p_nx_n^{a_n}=q_1y_1^{b_1}+...+q_my_m^{b_m} , has parametric
solution and infinitely many solutions in nonzero integers with the condition
that: there is a i such that p_i=1, and
(a_i,a_1....a_{i-1}a_{i+1}...b_1...b_m)=1, or there is a j such that q_j=1, and
(b_j,a_1...a_nb_1...b_{j-1}b_{j+1}..b_m)=1. Finally we study the DE
x^a_y^b=z^c.

Number Theory