Let P(w) be a polynomial and \(\lambda\) a number such that \(\lambda =P(\lambda)\), \(P'(\lambda)=1\) and \(P(w)=\lambda +(w-\lambda)+A_ m(w- \lambda)^{m+1}+...,\) (m\(\geq 1)\) then the difference equation \((1)\quad y(x+1)=P(y(x))\) possesses an entire solution \(\phi_{\lambda}(x).\) The author proves several theorems concerning entire solutions of (1). A typical result is the following Theorem: \(\phi_{\lambda}(x)\) cannot be factored with transcendental f and g, where g(x) satisfies \((2)\quad g(x+1)=G(g(x)),\) G entire. Here factorization is to be taken in the composition sense.