Thue's Theorem states that an algebraic irrational number of degree at least three can be approximated by rational numbers only to a limited degree, giving a bound on how closely a rational number with a given denominator can approach it that is stronger than the earlier bound proved by Liouville. Named for Axel Thue, it was a major advance in Diophantine approximation and was later improved further by Siegel and then by Roth, whose own theorem gives the essentially best possible bound.
Facts
Statementmu(alpha) <= d/2 + 1, where mu(alpha) is the irrationality exponent of the algebraic number alpha and d is its degree. 1 Classification
Statement Form Connections
Has Statement Form
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.
Proved By
Source Roth's Theorem (Thue-Siegel-Roth) (Wikipedia)
Sources
1. Thue-Siegel-Roth theorem, Wikipedia
History section, Thue's 1909 resultQuote, History section, Thue's 1909 result
and in Thue's theorem from 1909 established mu(alpha) <= d/2 + 1
Roth's Theorem (Thue-Siegel-Roth) (Wikipedia)
Wikimedia FoundationProved By: Carl Ludwig Siegel, Lead paragraphQuote, Proved By: Carl Ludwig Siegel, Lead paragraph
In mathematics, Roth's theorem or Thue-Siegel-Roth theorem is a fundamental result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating
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