Permutation binomials over finite fields

Abstract

We prove that if x m + a x n x^m + ax^n permutes the prime field F p \mathbb {F}_p , where m > n > 0 m>n>0 and a ∈ F p ∗ a\in \mathbb {F}_p^* , then gcd ( m − n , p − 1 ) > p − 1 \gcd (m-n,p-1) > \sqrt {p}-1 . Conversely, we prove that if q ≥ 4 q\ge 4 and m > n > 0 m>n>0 are fixed and satisfy gcd ( m − n , q − 1 ) > 2 q ( log ⁡ log ⁡ q ) / log ⁡ q \gcd (m-n,q-1) > 2q(\log \log q)/\log q , then there exist permutation binomials over F q \mathbb {F}_q of the form x m + a x n x^m + ax^n if and only if gcd ( m , n , q − 1 ) = 1 \gcd (m,n,q-1) = 1 .