Abstract. In this paper, we investigate the uniqueness problem on entire functionssharing fixed points (ignoring multiplicities). Our main results improve and generalizesome results due to Zhang [13], Qi-Yang [10] and Dou-Qi-Yang [1]. 1. IntroductionIn this paper, a meromorphic function will mean meromorphic in the wholecomplex plane. We assume that the reader is familiar with standard notations andfundamental results of Nevanlinna Theory as explained in [12].We say that two meromorphic functions f and g share a small function a(z) IM(ignoring multiplicities) when f −a and g−a have the same zeros. If f and g havethe same zeros with the same multiplicities, then we say that f and g share a(z)CM (counting multiplicities).Let p be a positive integer and a ∈ C. We denote by N p (r, 1f−a ) the countingfunction of the zeros of f − a where an m-fold zero is counted m times if m ≤ pand p times if m > p. We denote by N L (r, 1f−1 ) the counting function for 1-pointsof both f(z) and g(z) about which f(z) has a larger multiplicity than g(z), withmultiplicity not being counted. We say that a finite value z