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Faltings' theorem

A compact browser-based explainer built around three ARC parts: Answer, Reason, and Check.

Informal statement:
If a smooth projective curve has genus g ≥ 2 and is defined over a number field, then it has only finitely many rational points.

Answer

What the page is claiming

The curve y² = x(x+1)(x−2)(x+2)(x−3) is a genus‑2 curve. In the setting of Faltings' theorem, that places it in the regime where the set of rational points is finite.

Example curve

genus 2

Field in view

Theorem says

finite

Obvious rational points

genus 0: often infinite once parameterised genus 1: finite or infinite, with group law genus 2+: finite by theorem

This page uses computation to make the contrast believable, but the decisive conclusion in the genus‑2 case comes from the theorem itself, not from a finite search.

Reason

Why the theorem feels plausible when you compare the three regimes

Denominator bound

x-window for search

Multiples on genus 1

Genus 0 — the unit circle

Curve: x² + y² = 1. Once a rational point is known, there is a rational parametrisation.

real curverational points found

Found at current bound

–

Typical behaviour

many

Genus 1 — an elliptic curve

Curve: y² = x³ − 16x + 16. This sits in the elliptic-curve world, where rational points form a finitely generated abelian group.

real curverational points foundexact multiples of P=(0,4)

Found at current bound

–

Exact multiples shown

–

n	nP	Approximation

Genus 2 — the Faltings example

Curve: y² = x(x+1)(x−2)(x+2)(x−3). This is the regime where Faltings' theorem applies.

real curverational points found

Found at current bound

–

Theorem-level conclusion

finite

Growth comparison

This chart compares the number of rational points the browser finds as the denominator bound increases.

Check

Concrete browser-side verification

Check A — do the obvious points satisfy the genus‑2 equation?

Point	y²	x(x+1)(x−2)(x+2)(x−3)	Result

Check B — theorem premises in plain language

Check C — what the bounded search is and is not doing

The search box can support the story, but it cannot replace the theorem. A finite search never proves finiteness; it only checks sample behaviour inside a chosen window.