U-Bit Collapse in Arnault Composites:Probing the Boundary of Strong Lucas Pseudoprimes

TL;DR

This study investigates the behavior of certain composite numbers engineered to pass specific primality tests, revealing that they maintain strong independence between tests and exhibit minimal Lucas sequence collapse, thus supporting their reliability.

Contribution

The paper introduces the U-bit collapse metric and provides empirical evidence that composites resistant to Miller-Rabin tests also show negligible Lucas sequence degeneracy, reinforcing the robustness of combined primality tests.

Findings

Composite numbers show minimal Lucas collapse, with average delta around 1.6 bits.

26% of samples exhibit no measurable Lucas collapse.

Results support the independence of Miller-Rabin and Lucas tests in composite detection.

Abstract

We present a computational study of 200 composite integers of approximately 350 bits, engineered using the Arnault framework to pass all Miller-Rabin tests up to base 11. Generated at a rate of approximately 20 per hour from a high-throughput construction process producing ~7,700 Carmichael numbers per minute, all samples fail the strong Lucas probable prime test. We introduce the U-bit collapse metric delta = log_2(n) - log_2(U_d mod n) to quantify deviation from the expected uniform distribution of Lucas sequence terms. Analysis reveals minimal collapse values: mean delta = 1.61 bits, median delta = 1.0 bits, maximum delta = 8 bits, with 26% showing no measurable collapse. We analyze correlations with prime residue classes modulo 35, Arnault construction parameters (k,M), and composite bit-sizes. Our results demonstrate that composites engineered for Miller-Rabin resistance exhibit negligible Lucas sequence degeneracy, providing strong empirical evidence for the statistical independence of these two primality test components and supporting the continued robustness of Baillie-PSW-type tests.