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Let H denote the set of all the entire functions f(z) of the form: f(z) = h(z)eM + k(z) where p(z) is a nonconstant polynomial of degree m, and A(# 0), k (# constant) are two entire functions of order less than m. In this paper, a necessary and sufficient condition for a function in H to be a prime is established. Several generalizations of known results follow. Some sufficient conditions for primeness of various subclasses of H are derived. The methods used in the proofs are based ondoi:10.1090/s0002-9947-1974-0349972-3 fatcat:qxs5uggnpnhyjah4xgtfyrroxq