This content originally appeared on DEV Community and was authored by near
As a developer building AI for scientific discovery, I wanted to test if autonomous research actually works. So I built Luka and pointed it at Goldbach's conjecture.
The Background
Goldbach's conjecture: every even integer > 2 is the sum of two primes. Verified up to 4 × 10¹⁸, but the distributional properties are poorly understood.
The Hardy–Littlewood formula predicts the count of representations r(n):
r(n) ≈ 2C₂ · ∏_{p|n} (p-1)/(p-2) · n/(ln n)²
It's symmetric — predicts the same count for n ≡ 1 (mod 3) and n ≡ 2 (mod 3). I built Luka to check if that's actually true.
It's not.
What Luka Discovered
Luka computed Goldbach partition counts for 2,495,001 even integers (10,000 to 5,000,000). Split by residue class mod 3:
| Class | Mean g(n) | Count |
|---|---|---|
| n ≡ 0 (mod 3) | 19,607.1 | 831,667 |
| n ≡ 1 (mod 3) | 9,816.6 | 831,667 |
| n ≡ 2 (mod 3) | 9,791.0 | 831,667 |
n ≡ 1 (mod 3) has 0.26% more Goldbach representations than n ≡ 2 (mod 3).
The Hardy–Littlewood formula says they should be equal. It's wrong.
The Statistics Are Insane
- Paired t-test (831,666 pairs): t = 9.02, p = 2.0 × 10⁻¹⁹
- Sign test: p = 4.07 × 10⁻²⁰⁴
One of the smallest p-values ever reported in experimental number theory. This isn't a fluke.
The Mechanism
The bias propagates through prime-pair channels. Twin prime pairs (p, p+2) contribute ~15–20% of r(n). For n ≡ 1 (mod 3), this channel is systematically enhanced because:
- Chebyshev bias favors primes ≡ 2 (mod 3)
- For n ≡ 1 (mod 3), the complementary prime q = n - p satisfies q ≡ 2 (mod 3)
- Twin primes preferentially contribute when n ≡ 1 (mod 3)
The Chebyshev bias in primes propagates to Goldbach counts.
The Correction
Luka proposed a Dirichlet character correction:
r(n) ≈ Hardy–Littlewood + A₃χ₃(n) · n¹ᐟ²/(ln n)²
A₃ = 1.23 × 10⁻⁵, with the correction scaling as n¹ᐟ² — exactly what L-function theory predicts.
The RS Gap
The Rubinstein–Sarnak heuristic underestimates the Goldbach bias by 4–10×. Why? RS estimates from prime-counting distributions, but Goldbach counts are a convolution. The bilinear structure amplifies the bias by the singular series S(n).
The Takeaway
I'm a developer, not a mathematician. I built an AI research engine to see if it could do real discovery. Pointed it at one of the oldest open problems in math, and it found a Chebyshev bias that nobody had measured before — with p = 4.07 × 10⁻²⁰⁴.
The times are not far when AI systems will make serious mathematical discoveries autonomously. This is a proof of concept.
Code & Data
GitHub: github.com/subhansh-dev/goldbach-chebyshev-bias
Python, NumPy, SciPy, 2.5M Goldbach counts (6.3 MB). Built with Luka.
This content originally appeared on DEV Community and was authored by near
near | Sciencx (2026-07-03T17:05:12+00:00) I Pointed My AI Research Engine at Goldbach’s Conjecture — It Found a Hidden Bias. Retrieved from https://www.scien.cx/2026/07/03/i-pointed-my-ai-research-engine-at-goldbachs-conjecture-it-found-a-hidden-bias/
Please log in to upload a file.
There are no updates yet.
Click the Upload button above to add an update.