A Lock-Free, Fully GPU-Resident Architecture for the Verification of Goldbach's Conjecture

TL;DR

This paper introduces a fully GPU-resident, lock-free multi-GPU architecture that significantly accelerates the large-scale computational verification of Goldbach's conjecture, achieving near-zero host-device communication and high parallel efficiency.

Contribution

It presents a novel GPU-only pipeline with lock-free work-stealing and overflow guards, enabling faster and more scalable verification of Goldbach's conjecture compared to previous host-dependent methods.

Findings

Achieves 45.6x speedup over previous methods at N=10^{10}

Verifies Goldbach's conjecture up to 10^{12} in 36.5 seconds on a single GPU

Verifies up to 10^{13} in 133.5 seconds on four GPUs

Abstract

We present a fully device-resident, multi-GPU architecture for the large-scale computational verification of Goldbach's conjecture. In prior work, a segmented double-sieve eliminated monolithic VRAM bottlenecks but remained constrained by host-side sieve construction and PCIe transfer latency. In this work, we migrate the entire segment generation pipeline to the GPU using highly optimised L1 shared-memory tiling, achieving near-zero host-device communication during the critical verification path. To fully leverage heterogeneous multi-GPU clusters, we introduce an asynchronous, lock-free work-stealing pool that replaces static workload partitioning with atomic segment claiming, enabling $99.7$% parallel efficiency at 2 GPUs and $98.6$% at $4$ GPUs. We further implement strict mathematical overflow guards guaranteeing the soundness of the 64-bit verification pipeline up to its theoretical ceiling of $1.84 \times 10^{19}$. On the same hardware, the new architecture achieves a $45.6\times$ algorithmic speedup over its host-coupled predecessor at N = $10^{10}$. End-to-end, the framework verifies Goldbach's conjecture up to $10^{12}$ in $36.5$ seconds on a single NVIDIA RTX 5090, and up to $10^{13}$ in $133.5$ seconds on a four-GPU system. All code is open-source and reproducible on commodity hardware.