**Thue's Theorem** refers to several results by the Norwegian mathematician [[Axel Thue]], who made significant contributions to number theory and combinatorics. However, the term "Thue's Theorem" might refer to different theorems depending on the context. Here, I will discuss two notable theorems attributed to him: one in combinatorics related to square-free words and another in number theory related to Diophantine approximation. ### Thue's Theorem in Combinatorics In combinatorics, Thue is celebrated for his pioneering work on square-free and [[Overlap-free|overlap-free words]]. A [[square-free word]] is a sequence of symbols where no consecutive sequence of symbols (or substring) appears twice in a row. An overlap-free word is a sequence where no substring of the form $avava$ occurs, where $a$ is a single symbol, and $v$ is a sequence of symbols (possibly empty). **Thue's Theorem** states that infinite square-free words exist over a three-letter alphabet. This was a groundbreaking result in the early 20th century, showing that it's possible to construct sequences of symbols that avoid simple repetitive patterns, even with a limited set of symbols. Thue constructed such sequences using an iterative method now known as the [[Prouhet-Thue-Morse sequence]], although the sequence itself was originally defined to be [[overlap-free]]. ### Thue's Theorem in Number Theory In number theory, Axel Thue's work on Diophantine approximation led to what is now known as **Thue's Theorem**. This theorem concerns the solutions to certain algebraic equations and was an important development in the field. **Thue's Theorem** (1912) states that if $F(x,y)=0$ is an [[irreducibility|irreducible]] homogeneous polynomial in two variables with integer coefficients and degree at least $3$, then there are only finitely many integer pairs $(x,y)$ that satisfy the equation. This result was significant because it provided one of the first general results showing that higher-degree algebraic equations tend to have only finitely many integer solutions, a crucial insight for the development of modern Diophantine geometry. ### Impact and Legacy Both theorems have had a profound impact on their respective fields: - **In Combinatorics**: Thue's insights into non-repetitive sequences laid the groundwork for further research in combinatorial pattern avoidance, an area still active today with applications in computer science, especially in algorithm design and formal language theory. - **In Number Theory**: Thue's results on Diophantine approximation spurred further advancements, including the Thue-Siegel-Roth Theorem, which significantly strengthened Thue's original bounds on rational approximations to algebraic numbers. Thue's work exemplifies the deep connections between seemingly abstract mathematical theory and practical problems in algorithm design and number theory, highlighting the enduring value of foundational research. # References ```dataview Table title as Title, authors as Authors where contains(subject, "Thue's Theorem") sort modified desc, authors, title ```