Scientia Magna, December 1st, 2007
Abstract In his book "Comments and Topics on Smarandache Notions and Problems", Kenichiro Kashihara asked us to find all positive integer solutions of the Diophantine equations [([a.sup.b] x [b.sup.a]).sup.[2/a+b]] = c or [([a.sup.b] x [b.sup.c] x [c.sup.a]).sup.[3/a+b+c] = d. He also said, "The question I would like to ask is whether there exists positive integer solutions where a 6= b for either of the equations. If such non-trivial solutions exist, find a general method for generating them". In this paper, we using the elementary method to study these two problems, and prove that the firs...
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