A pseudoprime which obeys an additional restriction beyond that required for a Frobenius pseudoprime. A number n with (n, 2a) = 1 is a strong Frobenius pseudoprime with respect to x - a iff n is a strong pseudoprime with respect to f(x). Every strong Frobenius pseudoprime with respect to x - a is an Euler pseudoprime to the base a. Every strong Frobenius pseudoprime with respect to f(x) = x^2 - b x - c such that ((b^2 + 4c)/n) = - 1 is a strong Lucas pseudoprime with parameters (b, c). Every strong Frobenius pseudoprime n with respect to x^2 - b x + 1 is an extra strong Lucas pseudoprime to the base b.