Arithmetic Bias in Mersenne Prime Exponents and the Divisor Structure of $p-1$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917551875817472 |
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| author | Dominguez, Jesus |
| author_facet | Dominguez, Jesus |
| contents | According to the classical Wagstaff heuristic, the probability that a Mersenne number \(M_p=2^p-1\) is prime depends primarily on the size of the exponent \(p\). We investigate whether the divisor structure of \(p-1\) produces detectable secondary variations within this asymptotic framework. We introduce the normalized divisor parameter S(p)=\frac{\logτ(p-1)}{\log\log p}, which measures the divisor complexity of \(p-1\), including prime multiplicities. Using the currently known Mersenne prime exponents (excluding small cases), we compare \(S(p)\) against nearby prime controls of comparable size. Across several complementary distribution-free methods, including percentile analysis, conditional likelihood estimation, and permutation tests, Mersenne prime exponents consistently exhibit elevated values of \(S(p)\). To interpret this effect, we develop a heuristic framework based on the cyclotomic decomposition
2^{p-1}-1=\prod_{d\mid(p-1)}Φ_d(2), in which divisors of \(p-1\) generate effective modular constraint layers. This motivates a heuristic refinement of the Wagstaff model of the form
P(M_p\text{ prime}) \approx C\,\frac{(\log p)^{S(p)}}{p}. The proposed refinement preserves the classical Wagstaff scale in the typical regime \(S(p)\approx 1\), while suggesting that the finite-scale distribution of Mersenne prime exponents exhibits a weak arithmetic bias linked to the divisor structure of \(p-1\). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_08994 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Arithmetic Bias in Mersenne Prime Exponents and the Divisor Structure of $p-1$ Dominguez, Jesus Number Theory 11A41, 11N05, 11Y11, 62G10 According to the classical Wagstaff heuristic, the probability that a Mersenne number \(M_p=2^p-1\) is prime depends primarily on the size of the exponent \(p\). We investigate whether the divisor structure of \(p-1\) produces detectable secondary variations within this asymptotic framework. We introduce the normalized divisor parameter S(p)=\frac{\logτ(p-1)}{\log\log p}, which measures the divisor complexity of \(p-1\), including prime multiplicities. Using the currently known Mersenne prime exponents (excluding small cases), we compare \(S(p)\) against nearby prime controls of comparable size. Across several complementary distribution-free methods, including percentile analysis, conditional likelihood estimation, and permutation tests, Mersenne prime exponents consistently exhibit elevated values of \(S(p)\). To interpret this effect, we develop a heuristic framework based on the cyclotomic decomposition 2^{p-1}-1=\prod_{d\mid(p-1)}Φ_d(2), in which divisors of \(p-1\) generate effective modular constraint layers. This motivates a heuristic refinement of the Wagstaff model of the form P(M_p\text{ prime}) \approx C\,\frac{(\log p)^{S(p)}}{p}. The proposed refinement preserves the classical Wagstaff scale in the typical regime \(S(p)\approx 1\), while suggesting that the finite-scale distribution of Mersenne prime exponents exhibits a weak arithmetic bias linked to the divisor structure of \(p-1\). |
| title | Arithmetic Bias in Mersenne Prime Exponents and the Divisor Structure of $p-1$ |
| topic | Number Theory 11A41, 11N05, 11Y11, 62G10 |
| url | https://arxiv.org/abs/2603.08994 |