$D(-1)$-triples of triangular numbers

1. Problem Statement

The paper investigates Diophantine $m$-tuples within the specific subset of triangular numbers.

• Definition: A set of $m$ distinct positive integers is a $D(n)$-tuple if the product of any two distinct elements, increased by $n$, is a perfect square.
• Specific Focus: The author focuses on the property $D(-1)$, where the product of any two distinct elements minus 1 is a perfect square.
• Target Set: The elements must be triangular numbers, denoted as $T_k = \frac{k(k+1)}{2}$.
• Core Questions:
  1. Under what conditions does a triangular number $T_n$ belong to a $D(-1)$-pair $\{T_n, T_m\}$?
  2. If such a pair exists, can $T_n$ be extended to form a $D(-1)$-triple (and beyond) consisting entirely of triangular numbers?

2. Methodology

The author employs number-theoretic techniques, specifically the theory of generalized Pellian equations and linear recurrence sequences.

A. Transformation to Pellian Equations

The condition for a $D(-1)$-pair $\{T_n, T_m\}$ is:
$$T_n T_m - 1 = r^2$$
Substituting the formula for triangular numbers and performing algebraic manipulation transforms this into a generalized Pellian equation:
$$x^2 - n(n+1)y^2 = -(n(n+1) + 16)$$
where the variables are defined as:

• $x = 4r$
• $y = 2m + 1$

B. Analysis of Fundamental Solutions

The paper analyzes the solvability of this equation. It utilizes the fundamental solution of the corresponding standard Pell equation $X^2 - n(n+1)Y^2 = 1$, which is identified as $(X_1, Y_1) = (2n+1, 2)$.
The solutions to the generalized equation are generated via recurrence relations derived from the fundamental solution $(x^*, y^*)$ and the Pell equation's fundamental unit.

C. Parity and Modular Constraints

A rigorous analysis of modular arithmetic (modulo 4, 8, and prime factors) is conducted to determine necessary conditions for $n$ such that the equation has integer solutions where $y$ is odd and $x$ is divisible by 4.

3. Key Contributions and Results

Theorem 1: Necessary Conditions for Existence

The paper establishes a strict necessary condition for a triangular number $T_n$ to be part of a $D(-1)$-pair.

• Condition: Both $n$ and $n+1$ must be of the form $2^a \cdot p_1^{l_1} \cdots p_k^{l_k}$, where:
  • Every odd prime factor $p_i \equiv 1 \pmod 4$.
  • For $n$, the exponent $a \in \{0, 2\}$.
  • For $n+1$, the exponent $a \in \{0, 1\}$.
• Implication: This implies both $n$ and $n+1$ can be expressed as the sum of two squares.
• Counter-example: The author notes that while $n=100$ satisfies these conditions, it does not yield a valid $D(-1)$-pair with another triangular number, proving the condition is necessary but not sufficient.

Theorem 2: Infinite Extension to Triples

This is the central result of the paper.

• Statement: If a triangular number $T_n$ belongs to any $D(-1)$-pair $\{T_n, T_m\}$, then $T_n$ is a member of infinitely many $D(-1)$-triples consisting solely of triangular numbers.
• Mechanism: The author constructs a sequence of indices $(m_k)_{k \ge 1}$ using linear recurrence relations derived from the Pellian solutions. It is proven that for any $k \ge 1$, the set $\{T_n, T_{m_k}, T_{m_{k+1}}\}$ forms a valid $D(-1)$-triple.
• Proof Strategy: By showing that the product $T_{m_k}T_{m_{k+1}} - 1$ results in a perfect square through the identity:
  $$T_{m_k}T_{m_{k+1}} - 1 = \left( \frac{y_k y_{k+1} - (2n+1)}{8} \right)^2$$
  where $y_k = 2m_k + 1$.

Algorithmic Generation

The paper provides an algorithm to generate all indices $n$ up to a bound $C$ such that $T_n$ forms a $D(-1)$-pair.

• Example: For $C=1000$, the valid indices found are $\{1, 4, 25, 148, 457, 865\}$.
• Observation: If $T_m$ is in a pair, there exists a smaller $n < m$ such that $\{T_n, T_m\}$ is the pair, allowing for a recursive search starting from small seeds.

Theorem 3: Generalization to $D(a)$

The methodology is generalized for any non-zero integer $a$.

• Result: If $\{T_n, T_m\}$ is a $D(a)$-pair, then $T_n$ can be extended to infinitely many $D(a)$-triples of triangular numbers.
• Limitation: While the extension mechanism works generally, the existence of the initial pair depends heavily on the specific value of $a$, and the necessary conditions of Theorem 1 do not directly generalize.

4. Significance

• Resolution of a Specific Case: The paper resolves the structure of $D(-1)$-triples within triangular numbers, proving that the existence of a pair guarantees the existence of infinite triples.
• Methodological Advancement: It demonstrates how generalized Pellian equations can be effectively used to study Diophantine properties within specific integer sequences (triangular numbers), extending previous work on Fibonacci numbers and general Diophantine tuples.
• Foundation for Future Research: The paper highlights that while the extension to triples is universal for a given $a$, the criteria for the existence of the initial pair remain an open and complex problem dependent on the specific arithmetic properties of $a$. This sets a clear direction for future research into $D(a)$-tuples of triangular numbers.

In summary, the paper proves that for triangular numbers, the property of being part of a $D(-1)$-pair is a gateway to an infinite family of $D(-1)$-triples, provided the initial necessary number-theoretic conditions on the index $n$ are met.