Published October 9, 2025 | Version v1
Preprint Open
- 1. Black Star Labs
Description
This preprint presents a new, high-throughput engine for empirical verification of the even Goldbach conjecture at unprecedented scale. Leveraging a fixed “gear” of small primes and constant-residue search, the method reduces verification of even numbers to a single streaming pass with O(1) cost per value. The engine supports both segmented-sieve and 64-bit deterministic Miller-Rabin modes, and is capable of processing windows of up to a trillion even numbers in a matter of hours on a single workstation.
Goldbach states: every even number ≥ 4 is a sum of two primes. The naive check for an even n tries many primes p and hopes that n − p is prime. Our idea is simpler: fix a small set of pri…
Published October 9, 2025 | Version v1
Preprint Open
- 1. Black Star Labs
Description
This preprint presents a new, high-throughput engine for empirical verification of the even Goldbach conjecture at unprecedented scale. Leveraging a fixed “gear” of small primes and constant-residue search, the method reduces verification of even numbers to a single streaming pass with O(1) cost per value. The engine supports both segmented-sieve and 64-bit deterministic Miller-Rabin modes, and is capable of processing windows of up to a trillion even numbers in a matter of hours on a single workstation.
Goldbach states: every even number ≥ 4 is a sum of two primes. The naive check for an even n tries many primes p and hopes that n − p is prime. Our idea is simpler: fix a small set of primes Q = {q1, . . . , qK } (the “gear”), and for each even n only test p = n − q with q ∈ Q. If any p is prime, that even is verified. Empirically, one of the first few q works almost always, so the cost per n is nearly constant. We sweep large windows using a 1-bit-per-even coverage array. Primality is implemented either by sieving just the p-range needed (fast up to 1011) or by a deterministic 64-bit Miller–Rabin for slices near 1018.
The approach is implemented in C#/.NET, designed for parallel execution across CPUs, and produces checkpointed, auditable coverage data for each segment. All code, documentation, and result artifacts are fully open and reproducible (see linked repository). Results demonstrate universal coverage for K=300 (the first 300 primes as gear) with no observed exceptions at trillion-scale intervals. This work is intended as an auditable, practical tool for open mathematical exploration, not a proof.
This work was conducted independently at Black Star Labs without external funding. The implementation prioritizes reproducibility and accessibility, enabling verification on consumer hardware. All source code, implementation details, and result artifacts are freely available under Apache 2.0 license.
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Trillion_Scale_Fixed_Gear_Goldbach_Verification (1).pdf
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Additional details
2025-09-30
Date of software and engine creation.
2025-09-29
Date of initial mathematical formula creation.
- Oliveira e Silva, T., Herzog, S., & Pardi, S. (2014). Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4×10^18. Mathematics of Computation, 83(288), 2033-2060.
- Chen, J. R. (1973). On the representation of a large even integer as the sum of a prime and the product of at most two primes. Scientia Sinica, 16, 157-176.