🎯 Basic Goldbach Explorer
Goldbach's Conjecture
$$\forall n \in \mathbb{E}, n > 2: \exists p_1, p_2 \in \mathbb{P} \text{ such that } n = p_1 + p_2$$20
4
50
☄️ Goldbach's Comet
Goldbach Function & Hardy-Littlewood Estimate
$$G(n) = |\{(p_1, p_2) : p_1, p_2 \in \mathbb{P}, p_1 + p_2 = n, p_1 \leq p_2\}|$$ $$G(n) \sim 2\Pi_2 \prod_{p|n, p \geq 3} \frac{p-1}{p-2} \cdot \frac{n}{(\ln n)^2}$$1000
10
0
Average Representations
0
Maximum Representations
0
Points Plotted
0
Theoretical Average
⭕ Hardy-Littlewood Circle Method 3D
Circle Method on Complex Plane
$$S(\alpha) = \sum_{p \leq n} e^{2\pi i \alpha p}, \quad \int_0^1 S(\alpha)^2 e^{-2\pi i \alpha n} d\alpha$$30
Mouse: Drag=Rotate, Wheel=Zoom, Ctrl+Drag=Pan
Major Arcs (Blue): Near rational points a/q with small q
Minor Arcs (Green): Away from rationals
Amplitude Wave: Number of representations around perimeter
🧮 Chen's Theorem Visualization
Chen's Result (1973)
$$\forall n \text{ large, even}: \exists p \in \mathbb{P}, m \text{ such that } n = p + m$$where $m$ is prime or semiprime: $m = p_1 \cdot p_2$
100
🔬 Interactive Research Avenues
1. Advanced Circle Method
10
$$\left|\sum_{n \leq X} e^{2\pi i \alpha n}\right| \ll X^{1/2 + \epsilon}$$
2. Sieve Theory Refinements
5
$$S(\mathcal{A}, \mathcal{P}, z) = \sum_{n \in \mathcal{A}} \prod_{p \in \mathcal{P}, p < z} (1 - \chi_p(n))$$
3. L-functions and Zeros
50
$$L(s, \chi) = \sum_{n=1}^{\infty} \frac{\chi(n)}{n^s}$$