A note on the high power diophantine equations
farzali izadiProceedings - Mathematical Sciences
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Abstract
In this paper, we solve the simultaneous Diophantine equations (SDE) x µ 1 + x µ 2 + • • • + x µ n = k • (y µ 1 + y µ 2 + • • • + y µ n k), µ = 1, 3, where n ≥ 3, and k = n, is a divisor of n (n k ≥ 2), 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 µ = 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 n i=1 p i • x ai i = m j=1 q j • y bj j , 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 2 • • • a i−1 • a i+1 • • • a n • b 1 • b 2 • • • b m) = 1, or there is a j such that q j = 1, and (b j , a 1 • • • a n • b 1 • • • b j−1 • b j+1 • • • b m) = 1. Finally we study the DE x a + y b = z c .
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