# The Number of Permutation Binomials over ${\Bbb F}_{4p+1}$ where $p$ and $4p+1$ are Primes

### Abstract

We give a characterization of permutation polynomials over a finite field based on their coefficients, similar to Hermite's Criterion. Then, we use this result to obtain a formula for the total number of monic permutation binomials of degree less than $4p$ over ${\Bbb F}_{4p+1}$, where $p$ and $4p+1$ are primes, in terms of the numbers of three special types of permutation binomials. We also briefly discuss the case $q=2p+1$ with $p$ and $q$ primes.