Quadratic Frobenius pseudoprimes with respect to $x^{2}+5x+5$

Abstract

The quadratic Frobenius test is a primality test. Some composite numbers may pass the test and such numbers are called quadratic Frobenius pseudoprimes. No quadratic Frobenius pseudoprimes with respect to $x^{2}+5x+5$, which are congruent to 2 or 3 modulo 5, have been found. Shinohara studied a specific type of such a quadratic Frobenius pseudoprime, which is a product of distinct prime numbers $p$ and $q$. He showed experimentally that $p$ must be larger than $10^{9}$, if such a quadratic Frobenius pseudoprime exists. The present paper extends the lower bound of $p$ to $10^{11}$.