Diophantine m-tuples of Triangular Numbers

Abstract

A m-tuple with the property D(n) is a tuple of m positive integers (a₁, a₂, …, a_m) such that a_i a_j + n is an square, for 1 ≤ i < j ≤ m. The kth triangular number is T_k = (k(k+1))/2 for nonnegative integers k. We consider D(1) tuples consisting only of triangular numbers. We prove the nonexistence of any D(1) triangular quadruple and describe an algorithm to generate an infinite family of D(1) triangular triples, which we conjecture contains all D(1) triangular triples. We also consider general D(n) tuples. To aid with computational difficulties, we present an efficient algorithm, using Generalized Pell Equations (GPEs), to determine whether T_a is in a D(n) triangular pair, which runs in O(a^1/2) time. We then prove that no D(n) triangular pair exists for n ≡ 2,5 (mod 9), and discuss other values of n for which there appear to be no D(n) triangular pairs. We also show that our D(n) equation has solutions in all ℚ_p, for p ≠ 3. We then present progress on determining a general criteria on n for which no D(n) triangular pairs exist.

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BibTeX RIS

BibTeX

@misc{bagchi2026diophantine,
  title = {{Diophantine m-tuples of Triangular Numbers}},
  author = {Sounak Bagchi and Christian Zhou-Zheng},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.27697},
  eprint = {2608.27697},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.27697},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Diophantine m-tuples of Triangular Numbers
AU  - Sounak Bagchi
AU  - Christian Zhou-Zheng
PY  - 2026
DA  - 2026/08/27
PB  - arXiv
DO  - 10.48550/arxiv.2608.27697
UR  - https://arxiv.org/abs/2608.27697
AB  - A m-tuple with the property D(n) is a tuple of m positive integers (a_1, a_2, …, a_m) such that a_i a_j + n is an square, for 1 ≤ i < j ≤ m. The kth triangular number is T_k = (k(k+1))/2 for nonnegative integers k. We consider D(1) tuples consisting only of triangular numbers. We prove the nonexistence of any D(1) triangular quadruple and describe an algorithm to generate an infinite family of D(1) triangular triples, which we conjecture contains all D(1) triangular triples. We also consider general D(n) tuples. To aid with computational difficulties, we present an efficient algorithm, using Generalized Pell Equations (GPEs), to determine whether T_a is in a D(n) triangular pair, which runs in O(a^1/2) time. We then prove that no D(n) triangular pair exists for n ≡ 2,5 (mod 9), and discuss other values of n for which there appear to be no D(n) triangular pairs. We also show that our D(n) equation has solutions in all ℚ_p, for p ≠ 3. We then present progress on determining a general criteria on n for which no D(n) triangular pairs exist.
ER  -

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