Evaluation of the Effectiveness of the Frobenius Primality Test

The Frobenius primality test is based on the properties of the Frobenius automorphism of the quadratic extension of the residue field. Although it is probabilistic, we show that is "very rarely wrong". To date there are no counterexamples to this method and there are reasons to believe that they do not exist at all. In this paper, we suggest a version of the Frobenius test and prove that it does not fail for numbers less than $2^{64}$. We also show that a "Frobenius pseudoprime" will necessarily have a prime divisor greater than 3000.