Tiling a triangle into a prime number of congruent triangles

Technical Summary: Tiling a Triangle into a Prime Number of Congruent Triangles

Problem Statement
This paper addresses the problem of characterizing the integers $N$ for which a triangle $T$ can be dissected into $N$ congruent triangles $R$. Specifically, it investigates the conditions under which $N$ can be a prime number. The work builds upon previous research regarding Erdős problem 633 (concerning non-square tilings) and Erdős problem 634 (characterizing $N$ for which no tiling exists). The central question is: apart from known exceptions, can a triangle be tiled by a prime number $N > 3$ of congruent triangles?

Methodology
The author employs a case-based analysis grounded in the classification of tile shapes and the commensurability of angles and sides. The methodology proceeds as follows:

1. Classification of Cases: The analysis separates triangles $T$ based on whether they are equilateral, isosceles, or scalene, and whether their angles are commensurable (rational multiples of $\pi$) or incommensurable.
2. Leveraging Prior Results: The paper relies on established theorems (cited from works by Laczkovich, Beeson, and others) to eliminate cases where $T$ is equilateral, isosceles, or has commensurable angles. These prior results largely establish that for such triangles, $N$ is either not prime or restricted to specific small values ($N=2, 3$).
3. Focus on Incommensurable Angles: The core of the paper addresses the remaining difficult case: $T$ has incommensurable angles, is not equilateral or isosceles, and is tiled by a tile $R$ with angles $(\alpha, \beta, \gamma)$.
   • By Theorem 2, if $R$ has incommensurable angles and is not similar to $T$, $R$ must have commensurable sides. This allows the sides of $R$ to be treated as integers $(a, b, c)$.
   • The analysis focuses on "Group 2" tilings, defined by the condition $3\alpha + 3\beta = \pi$ (implying $\gamma = 2\pi/3$). Group 1 tilings ($3\alpha + 2\beta = \pi$) were previously resolved.
4. Algebraic Number Theory: For the four possible shapes of $T$ in Group 2, the author derives explicit formulas relating the number of tiles $N$ to the side lengths of the tile.
   • Using the Law of Sines and area equations, the side lengths of $T$ are expressed as linear combinations of the tile's sides.
   • The proportionality factor $\lambda$ between the sides of $T$ and a primitive integer triple is proven to be an integer (Lemma 14).
   • The area of $T$ is equated to $N$ times the area of the tile, yielding a factorization of $N$ in terms of $\lambda$ and the tile's side lengths.

Key Contributions and Results
The paper proves that if a triangle $T$ is tiled by $N$ congruent triangles $R$ (where $T$ is not similar to $R$), and $N > 3$, then $N$ cannot be a prime number.

The specific results for the four shapes of $T$ in the incommensurable Group 2 case are:

• Case 1: $T$ has angles $(\alpha, 2\alpha, 3\beta)$. The number of tiles is $N = 3\lambda^2(a + 2b)(a + b)$. Since $a, b, \lambda$ are positive integers, $N$ is composite.
• Case 2: $T$ has angles $(\alpha, 2\beta, 2\alpha + \beta)$. The number of tiles is $N = \lambda^2(2a + b)(a + b)$, which is composite.
• Case 3: $T$ has angles $(\alpha, \alpha + \beta, \alpha + 2\beta)$. The number of tiles is $N = \lambda^2(a + b)/b$. The author proves $a+b$ is composite (Lemma 15), ensuring $N$ is not prime.
• Case 4: $T$ has angles $(2\alpha, 2\beta, \alpha + \beta)$. The number of tiles is $N = \lambda^2(a + 2b)(2a + b)$, which is composite.

Significance and Main Theorem
The paper culminates in Theorem 22, which states: Let triangle $T$ be $N$-tiled by a tile not similar to $T$. Suppose $N > 3$. Then $N$ is not prime.

Combined with prior results on reptilings (tilings where the tile is similar to $T$) and the known exceptions (isosceles $N=2$, equilateral $N=3$, and specific right triangles), the paper provides a complete characterization of prime tilings in Corollary 23:

• An $N$-tiling of some triangle exists for a prime $N$ if and only if:
  • $N = 2$ (any isosceles triangle cut by altitude);
  • $N = 3$ (a 30-60-90 triangle);
  • $N \equiv 1 \pmod 4$ (a right triangle with legs in ratio $M/K$ where $N = M^2 + K^2$).

Consequently, the set of prime numbers $N$ for which no triangle can be tiled into $N$ congruent triangles consists exactly of those primes greater than 3 that are congruent to $3 \pmod 4$.

Modesty and Scope
The paper acknowledges that the "bulldozer-style" argument involves checking a finite number of cases. It notes that the number theory involved is simpler than that required for the related Erdős problem 633 (which dealt with non-square tilings and required elliptic equations), as the primary difficulty here lies in handling isosceles triangles, which were already resolved in prior literature. The paper also credits the discovery of Lemma 14 and Lemma 16 to an AI assistant (Claude Fable), highlighting the role of computational tools in verifying specific number-theoretic steps within the geometric proof.