On the Absence of Pseudoprimes in the Generalized Pell Primality Test of Bazzanella, Di Scala, Dutto & Murru (2022): A Structural Characterization and Algorithmic Improvement

This preprint addresses an open question posed by Bazzanella, Di Scala, Dutto and Murru (2022) regarding the Generalized Pell Primality Test. The authors verified empirically that no pseudoprimes exist up to 2⁶⁴ under Selfridge parameter selection but provided no structural explanation for this absence.

We present the first arithmetic characterization of this phenomenon. The central result is a necessary condition for a composite n = pq to be a pseudoprime of the test: the divisibility condition gcd(p+1, q−1) | (p−q) must hold, a requirement that fails for the overwhelming majority of semiprimes under Selfridge selection. An involution barrier argument is introduced to address the residual borderline case where gcd = 2, showing that such candidates impose factorization constraints incompatible with the pseudoprime condition.

Additionally, an algorithmic optimization of the test is presented using Non-Adjacent Form (NAF) exponent representation with a step-back formula for Lucas sequences, reducing non-zero digit density from approximately 1/2 to 1/3. A side-by-side computational comparison against the original binary Lucas fast-doubling algorithm is performed over all odd composites up to 10⁶, confirming zero pseudoprimes for both implementations and complete agreement between them, with a measured 5.6% reduction in modular multiplications.

The companion script implementing both algorithms is included as supplementary material.