Theorem

Siegel's Theorem on Integral Points

Number Theory

Siegel's Theorem on Integral Points states that an algebraic curve of genus at least one, defined over the rational numbers, has only finitely many points whose coordinates are both integers. Named for Carl Ludwig Siegel, who proved it in 1929, it is a foundational finiteness result of Diophantine geometry, a precursor to later results such as Faltings' Theorem, which strengthens the finiteness conclusion from integer points to all rational points on higher-genus curves.

Facts

Statement

For a smooth algebraic curve C of genus g defined over a number field K, presented in affine space in a given coordinate system, there are only finitely many points on C with coordinates in the ring of integers O of K, provided g is greater than zero. 1

Proof Year

1929 1

Classification

Statement Form

Existence Theorem 1

Statement Form

Inequality 1

Connections

Has Statement Form

Existence Theorem

Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.

Inequality

Proved By

Carl Ludwig Siegel

Source Siegel's theorem on integral points (Wikipedia)

Sources

1. Siegel's theorem on integral points (Wikipedia)

Statement section
For a smooth algebraic curve C of genus g defined over a number field K, presented in affine space in a given coordinate system, there are only finitely many points on C with coordinates in the ring of integers O of K, provided g > 0.
History section
In 1929, Siegel proved the theorem unconditionally by combining a version of the Thue-Siegel-Roth theorem, from diophantine approximation, with the Mordell-Weil theorem from diophantine geometry.
Proved By: Carl Ludwig Siegel, Lead paragraph
In mathematics, Siegel's theorem on integral points states that a curve of genus greater than zero has only finitely many integral points over

Switch Atlas

The Inner World

The Civil World

The Natural World

Siegel's Theorem on Integral Points

Has Statement Form

Proved By