  Permutation polynomials are an interesting subject of mathematics and have
applications in other areas of mathematics and engineering. In this paper, we
determine all permutation trinomials over $\mathbb{F}_{2^m}$ in Zieve's paper.
We prove a conjecture proposed by Gupta and Sharma and obtain some new
permutation trinomials over $\mathbb{F}_{2^m}$. Finally, we show that some
classes of permutation trinomials with parameters are QM equivalent to some
known permutation trinomials.
