The Mersenne Chain Reactor: A Physical Saturation Law for Mersenne Primes

Abstract The search for Mersenne primes at astronomical scales has reached a physical bottleneck: as numbers grow, the energy and time required for arithmetic verification become prohibitive on classical processors. Here we report a fundamental paradigm shift in the physics of computation and information. We introduce the Mersenne Chain Reactor (MCR), a physical system that determines the primality of Mersenne numbers without performing arithmetic operations. The reactor does not add, subtract, multiply, divide, or execute iterative numerical steps. Instead, primality is decided through a physical propagation process. By encoding the problem into a cyclic lattice and injecting a signal—optical, electrical, or quantum—the MCR transforms primality testing into a physical experiment. If the signal propagates and saturates the entire structure, the corresponding Mersenne number is prime; if destructive interference leaves persistent voids, it is composite. Crucially, the verdict is reached not by iterative computation, but by the completion of a single physical propagation event. The total latency is therefore bounded by the signal time-of-flight across the reactor geometry, yielding a constant physical depth independent of arithmetic complexity. In this sense, the MCR functions not as a calculator, but as an observational instrument—much as a balance does not compute weight, nor a thermometer compute temperature. We propose this architecture as a new experimental lens on the deep structure of prime numbers and a decisive shift from arithmetic computation to physical judgement.