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.
Categories: math.NT