Nonexistence of permutation binomials of certain shapes

TL;DR

This paper proves that certain binomials cannot be permutation polynomials over finite fields when specific gcd conditions are met, confirming a recent conjecture in special cases.

Contribution

It establishes new nonexistence results for permutation binomials of specific shapes over finite fields, extending previous conjectures.

Findings

gcd(m-n,p-1) is not 2 or 4 for such permutation binomials

Confirmed conjecture for cases where (p-1)/2 or (p-1)/4 is prime

Provides theoretical proof for nonexistence under given conditions

Abstract

Suppose x^m + c*x^n is a permutation polynomial over GF(p), where p>5 is prime, m>n>0, and c is in GF(p)^*. We prove that gcd(m-n,p-1) is not 2 or 4. In the special case that either (p-1)/2 or (p-1)/4 is prime, this was conjectured in a recent paper by Masuda, Panario and Wang.