Bayes Meets Riemann Again: Large Prime Discovery and Re-emergence of the Bone of Contention
Durba Bhattacharya, Sucharita Roy, Sourabh Bhattacharya
TL;DR
This work develops a Bayesian framework for prime distribution by modeling prime counts as a nonhomogeneous Poisson process with intensity Λ(t)=α∫_2^t li(u)du+β∫_2^t f(u)du, grounding the approach in the prime number theorem (PNT). A fast recursive Bayesian scheme updates stage-wise parameters (α_k,β_k) and uses Transformation-based MCMC (TMCMC) to sample from recursive posterior predictives, enabling discovery of very large primes and Mersenne-exponent candidates. The authors prove almost-sure consistency with the PNT, demonstrate asymptotic equivalence to non-recursive Bayesian inferences, and show robust falsification of the Riemann Hypothesis (RH) within this Bayesian framework, with RH failing under various model specifications. On the practical side, the TMCMC-based method identifies 259 primes beyond 140 million, including 184 strong Mersenne-prime exponents with digits ranging into the hundreds of millions, illustrating a scalable pathway for extreme-prime discovery. These results fuse probabilistic number theory with Bayesian computation to yield both theoretical insights and a concrete prime-hunting pipeline.
Abstract
Prime numbers have fascinated mathematicians since antiquity, with ongoing efforts to uncover both their properties and ever-larger examples. While giant primes rarely aid cryptography, they find use in areas such as locally decodable codes. Large prime-hunting, often brute-force in nature, is conceptually linked to the Riemann Hypothesis and the prime number theorem, which portrays prime distribution as essentially random. This motivates a statistical perspective, with Bayesian methodology providing a natural foundation. We show that the prime number theorem suggests a nonhomogeneous Poisson process for prime counts, yielding primes as waiting times. This process agrees with the prime number theorem, asymptotic results, and prime gap properties. Building on it, we develop a recursive Bayesian theory for large prime prediction and Riemann Hypothesis validation. The approach matches traditional but computationally infeasible non-recursive Bayesian formulations in the limit, and it strongly falsifies the Riemann Hypothesis. Finally, we propose a computational method using Transformation-based MCMC to simulate recursive posterior predictives. A simple change of variable enables simulation of Mersenne prime exponents. With modest computing resources, we identified 259 primes over 140 million, including 184 strong Mersenne candidates corresponding to potential primes with 42--242 million digits.