Abstract. In this paper, we investigate the high order difference coun-terpart of Bru¨ck’s conjecture, and we prove one result that for a tran-scendental entire function f of finite order, which has a Borel exceptionalfunction a whose order is less than one, if ∆ n f and f share one smallfunction d other than a CM, then f must be form of f(z) = a + ce βz ,where c and β are two nonzero constants such that d−∆ n ad−a = (e β −1) n .This result extends Chen’s result from the case of σ(d) < 1 to the generalcase of σ(d) < σ(f). 1. Introduction and main resultsIn this paper, a meromorphic function always means it is meromorphic inthe whole complex plane C. We assume that the reader is familiar with thestandard notations in the Nevanlinna theory. We use the following standardnotations in value distribution theory (see [8, 14, 15]):T(r,f), m(r,f), N(r,f), N(r,f),....And we denote by S(r,f) any quantity satisfyingS(r,f) = o{T(r,f)} as r → ∞,possibly outside of a set E with finite linear or logarithmic measure, not nec-essarily the same at each occurrence. A meromorphic function a(z) is said tobe a small function with respect to f(z) if and only if T(r,a) = S(r,f). Weuse λ(f) and σ(f) to denote the exponent of convergence of zeros of f and theorder of f respectively.We say that two meromorphic functions f(z) and g(z) share the value a IM(ignoring multiplicities) if f(z)−a and g(z)−a have the same zeros. If f(z)−aand g(z)−a have the same zeros with the same multiplicities, then we say thatthey share the value a CM (counting multiplicities). Classic Nevanlinna four