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In this paper we investigate the existence of permutation polynomials of the form x d + L(x) on $${{\mathbb{F}_{2^n}}}$$ , where $${{L(x)\in\mathbb{F}_{2^n}[x]}}$$ is a linearized polynomial. It is shown that for some special d with gcd(d, 2 n 驴1) 1, x d + L(x) is nerve a permutation on $${{\mathbb{F}_{2^n}}}$$ for any linearized polynomial $${{L(x)\in\mathbb{F}_{2^n}[x]}}$$ . For the Gold functions $${{x^{2^i+1}}}$$ , it is shown that $${{x^{2^i+1}+L(x)}}$$ is a permutation on $${{\mathbb{F}_{2^n}}}$$ if and only if n is odd and $${{L(x)=\alpha^{2^i}x+\alpha x^{2^i}}}$$ for some $${{\alpha\in\mathbb{F}_{2^n}^{*}}}$$ . We also disprove a conjecture in (Macchetti Addendum to on the generalized linear equivalence of functions over finite fields. Cryptology ePrint Archive, Report2004/347, 2004) in a very simple way. At last some interesting results concerning permutation polynomials of the form x 驴1 + L(x) are given.