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The number of permutation binomials over \({\mathbb F}_{4p+1}\) where \(p\) and \(4p+1\) are primes. (English) Zbl 1121.11077
Summary: 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 \({\mathbb F}_{4p+1}\), 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.

11T06 Polynomials over finite fields
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