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| Natura: | Preprint |
| Pubblicazione: |
2026
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| Accesso online: | https://arxiv.org/abs/2601.19817 |
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| _version_ | 1866912853534965760 |
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| author | Hall, Bowman |
| author_facet | Hall, Bowman |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_19817 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | U-Bit Collapse in Arnault Composites:Probing the Boundary of Strong Lucas Pseudoprimes Hall, Bowman Number Theory 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. |
| title | U-Bit Collapse in Arnault Composites:Probing the Boundary of Strong Lucas Pseudoprimes |
| topic | Number Theory |
| url | https://arxiv.org/abs/2601.19817 |