A note on the high power Diophantine equations

2017/05/11 by Baghalaghdam, Mehdi, Izadi, Farzali
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1705.04338

Abstract

In this paper, we solve the simultaneous Diophantine equations(SDE) x₁^u+...+x_n^u=k(y₁^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 p₁x₁^a₁+....+p_nx_n^a_n=q₁y₁^b₁+...+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,a₁....a_i-1a_i+1...b₁...b_m)=1, or there is a j such that q_j=1, and (b_j,a₁...a_nb₁...b_j-1b_j+1..b_m)=1. Finally we study the DE x^a_y^b=z^c.

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