Generalizations of the Fibonacci pseudoprimes test

Abstract

Di Porto and Filipponi recently described a generalization of the standard test for an odd composite integer n to be a pseudoprime (cf. [2]). Instead of evaluating powers of a given integer modulo n, they define a Fibonacci pseudoprime of the mth kind to be an odd composite integer n with the property

$$\text{V}{}_{\mathrm{n}}\text{(m) ≡ m}\text{mod}\text{n}$$

.

Here V_n(m) are the generalized Lucas numbers, or equivalently, the Dickson polynomials g_n(x; r) for r = −1 and evaluated at x = m. The Fibonacci pseudoprimes of the 1st kind are exactly the known Lucas pseudoprimes (cf. [16] and [18]). Here we consider several generalizations.