On Diophantine pairs and triples of triangular numbers

Marija Bliznac Trebje\v{s}anin

On Diophantine pairs and triples of triangular numbers

T0 review · 2 major / 1 minor · reviewed 2026-07-13 · grok-4.5

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Pith's one-line read Any triangular number already in a D(a)-pair extends to infinitely many D(a)-triples of triangular numbers.

desk verdict Abstract-only: clean-looking extension theorems for triangular D(a)-pairs/triples inside a specialized literature; proofs invisible so confidence stays low. read the letter →

arxiv 2603.29565 v2 pith:A42NDNOG submitted 2026-03-31 math.NT

Marija Bliznac Trebješanin This is my paper · ORCID

classification math.NT MSC 11D0911B83

keywords DiophantinepairstriplestriangularnumbersD(a)-setsnumbertheoryequations

topics The Collatz Conjecture

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies Diophantine pairs and triples made entirely of triangular numbers that satisfy the D(a) property for a fixed nonzero integer a: the product of any two distinct members, increased by a, is a perfect square. Its central claim is that membership of a triangular number in such a pair is enough to guarantee that the same number can be completed to a D(a)-triple of triangular numbers in infinitely many distinct ways. The result therefore turns a finite local condition into an infinite supply of larger configurations. In addition the author produces infinite families of integers a for which D(a)-pairs of triangular numbers exist, and other infinite families for which no such pairs can exist at all. A sympathetic reader cares because the statement organises the landscape of which a are admissible and shows that once a single pair appears, the set of triangular triples for that a is automatically infinite.

What carries the argument

The D(a)-property for triangular numbers: two (respectively three) distinct triangular numbers form a D(a)-pair (triple) when the product of any two of them, increased by the fixed nonzero integer a, is a perfect square. The argument uses this algebraic condition to produce an infinite parametric family of third triangular numbers once a pair is given.

What would settle it

Exhibit a concrete nonzero integer a and a concrete triangular number T that forms a D(a)-pair with some other triangular number, yet for which only finitely many (or zero) triangular numbers S exist such that {T,S,U} is a D(a)-triple of triangular numbers for some U.

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Extended reading notes

Core claim

If a triangular number belongs to a D(a)-pair of triangular numbers, then it can be extended to infinitely many distinct D(a)-triples consisting entirely of triangular numbers. The same work also exhibits infinite families of a that admit such pairs and infinite families of a that admit none.

Load-bearing premise

That there is a uniform constructive procedure which, given any triangular number already sitting in a D(a)-pair, manufactures infinitely many distinct triangular numbers that complete a D(a)-triple.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The one thing worth knowing is the central claim: any triangular number that already sits in a D(a)-pair extends to infinitely many D(a)-triples that are also triangular, plus infinite families of a that admit such pairs and families that forbid them. That is the whole paper in one sentence.

What is actually new is those three existence/classification statements inside the established Diophantine m-tuple program restricted to triangular numbers. The extension result is the strongest piece; the families of a are the usual parametric work. On the face of the abstract the claims are precise, of standard form for the subfield, and free of circularity or data-fitting. No broader reorganization of number theory is promised, which is honest.

The soft spot is purely informational and proportionate: we have only the abstract, so the uniform constructive procedure (the map or recurrence that produces the infinitely many third triangular numbers) is not exhibited. The reader and stress-test both flag this correctly as a gap, not as a detected flaw. There is no visible internal contradiction with known Pell-type methods or triangular-number identities, and nothing suggests the argument collapses under its own equations. If the full text supplies the usual recurrences and checks the product-plus-a condition stays triangular, the result will stand as solid specialized progress; if not, the referee will catch it.

This is for people already working on Diophantine m-tuples or figurate numbers. A specialist gets clear statements and (presumably) constructions; outsiders will not. It deserves a serious referee rather than desk rejection—the claims are sharp enough and the literature expects exactly this kind of theorem. I would send it out.

Referee Report

2 major / 1 minor

Summary. The manuscript studies Diophantine pairs and triples of triangular numbers with the property D(a) for a nonzero integer a. From the abstract, the principal claims are that any triangular number already belonging to a D(a)-pair extends to infinitely many D(a)-triples of triangular numbers; that there exist infinite families of integers a admitting such pairs; and that there exist families of a for which no triangular D(a)-pairs exist.

Significance. If established, the extension result would supply a constructive infinitude statement for triangular D(a)-triples, in the spirit of known results for other polygonal sequences in the Diophantine m-tuple literature. The accompanying existence and non-existence families for a would clarify the arithmetic constraints under which triangular D(a)-pairs can occur, and would therefore be of interest to specialists working on Diophantine m-tuples restricted to special number sequences.

major comments (2)

[Abstract] Only the abstract is available for review. The central claim—that membership of a triangular number in a D(a)-pair yields infinitely many triangular D(a)-triples—is load-bearing and is asserted without exhibition of the construction, recurrence, or arithmetic conditions that guarantee the product-plus-a condition remains triangular. Without the full text (lemmas, equations, or explicit maps), the claim cannot be verified or refuted, so no soundness determination is possible.
[Abstract] The abstract likewise asserts infinite families of a that admit triangular D(a)-pairs and families for which none exist, again without stating the parametric forms or the Diophantine conditions used to produce them. These classification statements are essential to the paper’s second and third claims and cannot be assessed from the abstract alone.

minor comments (1)

[Abstract] The abstract is clear and conventional in form, but a full manuscript would need to supply explicit notation for triangular numbers, the precise definition of a D(a)-pair/triple used, and at least one concrete numerical example of an extension.

Circularity Check

0 steps flagged · score 0.0 of 10

Abstract-only pure-math claims; no circularity detectable from available text.

full rationale

Only the abstract is available. It asserts standard existence and non-existence results for Diophantine D(a)-pairs and triples of triangular numbers: that any triangular number already in a D(a)-pair extends to infinitely many triangular D(a)-triples, plus infinite families of a that admit such pairs and families that admit none. There are no equations, fitted parameters, self-citations, uniqueness theorems, or ansatzes visible. Nothing can be reduced by construction to its inputs. The abstract-only setting precludes locating self-definitional loops or fitted-input-as-prediction patterns. Per the hard rules, an honest non-finding is required: score 0, empty steps. The reader's residual concern about an unexhibited constructive map is an information gap, not circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

Abstract-only review. No free parameters or invented entities are visible. Background axioms are the standard definitions of triangular numbers and of Diophantine pairs/triples with property D(a) for nonzero integer a; these are domain assumptions of the subfield, not ad-hoc inventions of the paper.

assumptions (3)

standard math Triangular numbers are numbers of the form n(n+1)/2 for integer n ≥ 1.
Standard definition assumed throughout; not proved in the abstract.
domain assumption A set {x1,...,xk} has property D(a) if xi*xj + a is a perfect square for all i ≠ j, with a a fixed nonzero integer.
Standard definition of Diophantine m-tuples with property D(a); the paper specializes it to triangular numbers.
domain assumption a is a fixed nonzero integer.
Stated in the abstract as the ambient setting for all claims.

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Cite this review

Pith. "Pith review of On Diophantine pairs and triples of triangular numbers." pith.science (2026). https://pith.science/paper/A42NDNOG

@misc{pith2026260329565,
  author       = {Pith},
  title        = {Pith review of: On Diophantine pairs and triples of triangular numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A42NDNOG}},
  note         = {Machine review of arXiv:2603.29565}
}

abstract

We investigate Diophantine pairs and triples of triangular numbers with the property $D(a)$ for a non-zero integer $a$. We prove that if a triangular number belongs to a $D(a)$-pair, it can be extended to infinitely many $D(a)$-triples of triangular numbers. Additionally, we determine infinite families of integers $a$ that admit such pairs, as well as families for which no $D(a)$-pairs can exist.

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Reviewed July 13, 2026 · model on record in the stance chip above.