JP Journal of Algebra, Number Theory and Applications
Volume 11, Issue 2, Pages 209 - 221
(August 2008)
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CHARACTERIZING THE SOLUTIONS OF A CERTAIN EXPONENTIAL DIOPHANTINE EQUATION
Stephen M. Zemyan (U. S. A.)
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Abstract: For a given odd integer I, it has been shown that the exponential Diophantine equation
has at most one solution, M and N, in integers, except when
5 or 13. In this paper, we show that many values of I are omitted, and derive necessary conditions on M and N for
to be assumed. In general, we find that when
is prime, then either M and N are relatively prime or M and N are bothsmall multiples of the same odd prime. We also study the divisibility properties of the assumed values of the equation. In particular, we show that if I is an assumed value of the equation and where q is prime and then M and N must have certain forms depending upon q and a. |
Keywords and phrases: exponential Diophantine equations, congruences, primitive roots. |
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