The distribution of Lucas and elliptic pseudoprimes

Abstract

Let L ( x ) \mathcal {L}(x) denote the counting function for Lucas pseudoprimes, and E ( x ) \mathcal {E}(x) denote the elliptic pseudoprime counting function. We prove that, for large x, L ( x ) ≤ x L ( x ) − 1 / 2 \mathcal {L}(x) \leq xL{(x)^{ - 1/2}} and E ( x ) ≤ x L ( x ) − 1 / 3 \mathcal {E}(x) \leq xL{(x)^{ - 1/3}} , where \[ L ( x ) = exp ⁡ ( log ⁡ x log ⁡ log ⁡ log ⁡ x / log ⁡ log ⁡ x ) . L(x) = \exp (\log x\log \log \log x/\log \log x). \]