সংরক্ষণ করুন:
| প্রধান লেখক: | |
|---|---|
| বিন্যাস: | Preprint |
| প্রকাশিত: |
2026
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| বিষয়গুলি: | |
| অনলাইন ব্যবহার করুন: | https://arxiv.org/abs/2603.08994 |
| ট্যাগগুলো: |
ট্যাগ যুক্ত করুন
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সূচিপত্রের সারণি:
- 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 \(p-1\).