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# Goldbach Conjecture Explorer

[GitHub](https://github.com/HectorTablero/QED-Scripts/blob/main/Magazine%201/Uncle%20Petros%20and%20Goldbach's%20Conjecture/goldbachExplorer.js) (Open Source, MIT License) | [Live Demo](https://qed.mat.uam.es/revista/articulo/petros-goldbach)

**Date:** July 2026
**Technologies:** JavaScript, Tailwind CSS

---

## Overview

GoldbachExplorer reproduces the central tension of the book review it accompanies: Goldbach's conjecture,

$$\forall\, n \in 2\mathbb{Z},\ n > 2, \quad \exists\ p, q \text{ prime} : n = p + q,$$

looks like a five-minute exercise, which is exactly why Uncle Petros hands it to his nephew as a summer puzzle. The element shows both halves of that trap side by side: an arc diagram of every prime pair summing to a chosen $n$ (never scarce), and a scatter plot of the representation count $g(n) = \#\{(p,q) : p \le q,\ p+q=n,\ p,q \text{ prime}\}$ climbing away from zero as $n$ grows (the "Goldbach comet"), which is precisely why no amount of brute-force checking will ever settle the conjecture: verifying $g(n) \ge 1$ for every even $n$ up to some bound, however large, is a statement about finitely many numbers, while the conjecture quantifies over all of them.

```javascript
// Example element configuration: [n, limit]
::: js GoldbachExplorer
100, 4000
goldbach1
autoPlay, allowPause, autoPauseOnScroll, allowFullscreen, openControls, h=200
:::
```

### Implementation

**Sieve and pair search.** `primesUpTo()` builds a simple Eratosthenes sieve up to `limit + 10`, running in $O(m \log\log m)$ for a bound $m$. `pairsFor(n)` walks primes $p \le n/2$ and checks whether $n - p$ is also prime, collecting every representation of $n$ as a sum of two primes — an $O(\pi(n/2))$ search per $n$ once the sieve exists, where $\pi$ is the prime-counting function.

**Arc diagram.** Each prime pair $(a, b)$ summing to $n$ is drawn as a semicircular arc over the number line, with arc height staggered linearly across the $k$-th of $m$ pairs as $h_k = 0.18 + 0.8\,\dfrac{k}{m-1}$, purely so that overlapping arcs for the same $n$ remain visually distinguishable rather than stacking on top of each other.

**Incremental comet fill.** Computing representation counts for every even number up to `limit` (default 4000) in one pass would stall the UI, so `main()` — the per-tick update hook — processes a bounded slice of 40 numbers per call, accumulating into an `Int32Array` and re-invalidating the canvas after each slice, so the comet visibly grows rather than appearing instantly. A `filling` flag (deliberately distinct from the runtime's own `running` property) tracks whether the sweep has finished, and the "draw the comet" button resets it to restart the fill.

### Technical Considerations

The slider is constrained to even values only (`2 * Math.round(v / 2)`), since odd $n$ has no bearing on the conjecture, and is clamped to `[4, limit]` regardless of how the value arrives. The comet plot's per-point drawing bypasses higher-level plotting helpers in favor of direct `fillRect` calls sized at 1.4px, since it needs to place potentially thousands of points per frame. The current slider position is marked on the comet with a colour (`GoldbachExplorer.MARK`) chosen to read as part of the site's palette rather than as an alert, verified against the actual rendered plume colour to clear a 5:1 contrast ratio in both light and dark themes.