On extensions of $D(4)$-triples by adjoining smaller elements

1. Problem Statement

The paper investigates the properties of Diophantine $m$-tuples, specifically sets of distinct positive integers $\{x_1, \dots, x_m\}$ such that the product of any two distinct elements plus 4 is a perfect square (property $D(4)$).

The central problem addressed is the uniqueness of extensions for $D(4)$-triples.

• Context: It is known that any $D(4)$-triple $\{a, b, c\}$ can be extended to a "regular" quadruple $\{a, b, c, d_+\}$ where $d_+ > \max\{a, b, c\}$.
• The Conjecture: It is conjectured that this extension is unique for elements larger than the triple elements. However, the paper focuses on Conjecture 1.2: If $\{a_1, b, c, d\}$ is a $D(4)$-quadruple with $a_1 < b < c < d$, can there exist a different integer $a_2$ (where $a_1 \neq a_2 < b$) such that $\{a_2, b, c, d\}$ is also a $D(4)$-quadruple?
• Goal: The authors aim to prove that such a scenario is impossible or, if possible, to establish strict constraints on the elements, ultimately showing that only finitely many such counterexamples exist.

2. Methodology

The authors employ a combination of algebraic number theory, Pellian equations, and the hypergeometric method (Diophantine approximation).

• System of Pellian Equations:
  Assuming the existence of two distinct extensions $\{a_1, b, c, d\}$ and $\{a_2, b, c, d\}$ with $a_1 < a_2 < b < c < d$, the authors define integers $r_i, s_i, t, x_i, y, z$ satisfying specific quadratic relations (e.g., $a_1b+4=r_1^2$, $bc+4=t^2$). This leads to a system of simultaneous Pellian equations:
  $$a_1z^2 - cx_1^2 = 4(a_1-c)$$
  $$a_2z^2 - cx_2^2 = 4(a_2-c)$$
  $$bz^2 - cy^2 = 4(b-c)$$
  Solutions to these equations are expressed via recurrence sequences $v_m$ and $w_n$. The problem reduces to finding indices $m, n$ such that $v_m = w_n = z$.

• Regularity vs. Irregularity:
  The authors distinguish between "regular" quadruples (generated by the standard formula $d_+$) and "irregular" ones. They prove that if two such quadruples exist sharing the same $\{b, c, d\}$, both must be irregular. This is a crucial step because irregular quadruples satisfy different bounds and properties than regular ones.

• Bounds and Approximation:

  • Lower Bounds: Using results from previous literature (specifically [2] and [10]), they establish lower bounds for the indices $m$ and $n$ in terms of $b$ and $c$ (e.g., $n > 0.09226 b^{-1/2} c^{1/4}$).
  • Upper Bounds: They utilize Theorem 2.2 (a result on linear forms in logarithms) to derive an upper bound for the index $n$ based on the size of the elements $a_1, a_2, b, c$.
  • Contradiction: By comparing the lower and upper bounds for $n$, they derive inequalities that restrict the possible values of $a_1, a_2, b,$ and $c$.

• Computer Search:
  For cases where analytical bounds are not sufficient to rule out solutions entirely, the authors use Wolfram Mathematica to perform exhaustive searches within the derived finite ranges for $a_1, a_2, b,$ and $c$.

3. Key Contributions and Results

Main Theorem (Theorem 1.3)

The paper establishes strict necessary conditions for the existence of two distinct $D(4)$-quadruples $\{a_1, b, c, d\}$ and $\{a_2, b, c, d\}$ with $a_1 < a_2 < b < c < d$:

1. Gap Constraint: $a_2 > 4a_1$.
2. Quadratic Constraint: $a_2 > 0.0251 a_1^2$.
3. Magnitude Constraints: If $a_1 \ge 2$, then $b < a_2^2$ and $a_2 \ge 317$.
4. Cubic Constraint: $0.25 a_1^2 b^3 < c < 0.25 a_2^2 b^3$.

Corollaries and Implications

• Corollary 1.4 (Irregularity): If such a pair of quadruples exists, both must be irregular. This implies that if the conjecture of unique regular extensions holds (Conjecture 1.1), then Conjecture 1.2 (uniqueness of smaller extensions) must also hold.
• Corollary 1.5 (Limit on Extensions): For any $D(4)$-triple $\{b, c, d\}$, there are at most two positive integers $a < \min\{b, c, d\}$ that can extend it to a $D(4)$-quadruple.
• Corollary 1.6 (Finiteness): There are only finitely many quintuples $\{a_1, a_2, b, c, d\}$ satisfying the counterexample conditions. This is proven by establishing an upper bound for $c$ ($c < 10^{2157}$) in irregular quadruples, reducing the search space to a finite set.

Specific Bounds Proven

• Section 4: Proves $c < 0.25 a_2^2 b^3$ and $c > 0.25 a_1^2 b^3$.
• Section 5: Proves $a_2 > 4a_1$ and $a_2 > 0.0251 a_1^2$.
• Section 6: Proves $b < a_2^2$ for $a_1 \ge 2$.

4. Significance

• Advancement over Previous Work: The paper significantly improves upon the authors' previous work (specifically reference [3]) by providing much tighter relations between the elements of hypothetical counterexamples.
• Reduction of Search Space: By proving that $a_2$ must be significantly larger than $a_1$ and that $b$ is bounded by $a_2^2$, the authors drastically reduce the number of cases that need to be checked computationally.
• Support for Uniqueness Conjectures: While not fully proving Conjecture 1.2 (which would require showing no such quadruples exist), the paper proves that any counterexample must be extremely rare (finite in number) and must satisfy very specific, restrictive algebraic conditions.
• Methodological Rigor: The paper demonstrates the effective application of the hypergeometric method combined with computer verification to solve complex Diophantine problems involving $D(n)$-tuples.

Conclusion

The paper establishes that the existence of two distinct $D(4)$-quadruples sharing the same three largest elements is highly constrained. It proves that such configurations, if they exist, must be irregular and satisfy strict inequalities ($a_2 > 4a_1$, $b < a_2^2$, etc.). Ultimately, it reduces the problem to a finite search, proving that there are only finitely many such "counterexamples," thereby strengthening the evidence for the uniqueness of Diophantine extensions.