  The subject matter of this work is the diophantine equation
x^n+y^m=c(x^k)(y^l), where n,m,k,l,c are natural numbers.We investigate this
equation from the point of view of positive integer solutions.A preliminary
examination of sources such as reference[1](L.E.Dickson's History of the Theory
of Numbers, Vol.II) and [2](W.Sierpinski's Elementary Theory of Numbers) shows
that little or no material can be found regarding this diophantine
equation.Note that when c=1, (x,y)=(1,1) is a solution regardless of the values
of the exponents n,m,k,and l. In Section3, five results from number theory are
listed.The first four are well known and are stated without proof.Result5 is of
central importance and it is used in the proofs of most of the nine theorems of
this paper.We offer a detailed proof of Result5. The entire paper is organized
according to eight cases. Here is a sample of two of the nine theorems. In
Theorem2, we prove that if n<or=k and n<m<k+l, then the above has no positive
integer solutions if c is not equal to 2;if c=2, then the above equation has
the unique solution(x,y)=(1,1). In Theorem5, part(vi), we prove that if c=3,
n-k=1,l=1,and m-n=1.Then the above equation has exactly two solutions:the
(x,y)=(2,2),(2^k+1,2^k).
