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

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.

💡 Research Summary

The paper presents a large‑scale computational experiment aimed at probing the relationship between Miller–Rabin (MR) resistance and strong Lucas pseudoprime resistance. Using the Arnault construction framework, the author generated composite numbers of roughly 350 bits that are guaranteed to pass MR tests for bases 2 through 11. The generation pipeline runs on a single‑core ARM Graviton EC2 instance and produces about 7,700 Carmichael numbers per minute. From this torrent, only about 0.015 % (≈20 per hour) satisfy the MR‑base‑11 criteria; a total of 200 such composites were collected for detailed study.

Each composite n is of the form p₁·p₂·p₃, where the primes are chosen so that pᵢ − 1 share a large common factor f together with a few small co‑factors. This design forces the MR condition “base d divides pᵢ − 1” to hold for all three primes, ensuring MR‑base‑11 success. For each n the strong Lucas test is applied using the standard discriminant selection (the first D with (D/n) = −1). The test checks whether U_d ≡ 0 (mod n) or V_{d·2^r} ≡ 0 (mod n) for some r < s, where n + 1 = d·2^s with d odd. All 200 composites failed the strong Lucas test; no strong Lucas pseudoprime was found.

To quantify how far the Lucas term U_d deviates from the uniform distribution expected for a random composite, the author introduces the “U‑bit collapse” metric
δ(n,D) = log₂ n − log₂ (U_d mod n).
If U_d were uniformly random in