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arxiv: 2604.17018 · v2 · submitted 2026-04-18 · 🧮 math.NT

On Regular Higher Power Rational Diophantine Triples

Alen Andra\v{s}ek This is my paper

Pith reviewed 2026-05-10 06:38 UTC · model grok-4.3

classification 🧮 math.NT
keywords rational Diophantine triplesfourth powersDiophantine equationsinfinite familiespositive rationalsnumber theoryhigher powers
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The pith

There are infinitely many triples of distinct positive rationals where each pair product plus one is a fourth power.

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

The paper constructs infinite families of rational Diophantine triples in which the product of any two plus one is a fourth power of a rational. It establishes that infinitely many such triples can have all three elements positive. This matters because it shows the higher-power version of the Diophantine tuple property is possible over the positive rationals in unbounded supply. The constructions use parametric choices of rationals that satisfy the required power conditions simultaneously. The work also identifies obstacles when the same approach is tried for sixth and eighth powers.

Core claim

We show that there are infinitely many triples with positive elements for k=4 by producing non-trivial infinite families of distinct nonzero rational numbers a, b, c such that ab + 1, ac + 1 and bc + 1 are all fourth powers.

What carries the argument

Parametric families of rational numbers satisfying the simultaneous conditions that each pairwise product plus one equals a rational fourth power.

If this is right

  • Infinitely many positive rational triples exist for the fourth-power case.
  • The parametric method produces non-degenerate families for k=4.
  • The same construction technique meets concrete obstacles when applied to sixth and eighth powers.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The families might be used to test whether four-element sets with the fourth-power property are possible.
  • Explicit numerical members of the families could be checked to see how small the positive examples can be made.
  • The difficulties noted for k=6 and k=8 suggest that even exponents may require different parametrizations beyond a certain point.

Load-bearing premise

The constructions assume the existence of suitable rational parameters satisfying the fourth-power condition without introducing contradictions or degeneracies in the rational numbers.

What would settle it

Explicit computation of the first parameter set in one of the constructed families, followed by direct verification that the three pairwise products plus one are all perfect fourth powers of rationals (or demonstration that every such parameter choice forces a repeated or zero element).

read the original abstract

A rational Diophantine $m$-tuple is a set $\{a_1,\ldots,a_m\}$ of distinct nonzero rational numbers such that $a_i a_j+1$ is a square for all $1\leq i < j\leq m$. Similarly, we may ask when $a_ia_j+1$ is a $k$-th power. Here, we study the case $k=4$ and produce some non-trivial infinite families of such triples. We show that there are infinitely many triples with positive elements for $k=4$. We also briefly consider the $k=6$ (sextic) and $k=8$ (octic) cases, explaining the difficulties in extending the method to higher exponents.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript constructs non-trivial infinite families of rational Diophantine triples {a, b, c} such that ab + 1, ac + 1, and bc + 1 are fourth powers. It proves there are infinitely many such triples consisting of positive rational numbers and briefly discusses obstructions to extending the method to sixth and eighth powers.

Significance. If the explicit parametric families are correct and the positivity argument is complete, the work supplies concrete infinite families for the quartic case, which is a substantive addition to the literature on higher-power Diophantine tuples. The explicit constructions themselves constitute a strength, as they allow direct generation of examples rather than relying solely on existence via rank arguments.

major comments (1)
  1. [main construction of the positive family] The proof that there are infinitely many positive triples (the central claim) must establish that infinitely many rational points on the auxiliary variety produce a, b, c > 0 simultaneously. Sign patterns or growth of denominators can restrict positivity to a finite subset even when the set of rational points is infinite; the manuscript needs an explicit infinite subfamily, a cone argument, or a height-based density statement to confirm this.
minor comments (2)
  1. [Abstract] The abstract states that 'some non-trivial infinite families' are produced; a sentence indicating the number of independent parameters or the dimension of the family would improve clarity.
  2. [Section 2] Notation for the auxiliary curves or surfaces used in the parametrization should be introduced once and used consistently; occasional shifts between 'curve' and 'variety' are mildly confusing.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying a point that requires clarification in the positivity argument. We address the major comment below and will incorporate the suggested strengthening into the revised manuscript.

read point-by-point responses

  1. Referee: [main construction of the positive family] The proof that there are infinitely many positive triples (the central claim) must establish that infinitely many rational points on the auxiliary variety produce a, b, c > 0 simultaneously. Sign patterns or growth of denominators can restrict positivity to a finite subset even when the set of rational points is infinite; the manuscript needs an explicit infinite subfamily, a cone argument, or a height-based density statement to confirm this.

    Authors: We agree that an explicit verification of positivity for an infinite subfamily strengthens the central claim. Our construction proceeds via an explicit rational parametrization of the auxiliary variety by a single rational parameter t. For all sufficiently large positive rational values of t (e.g., the infinite sequence t = n for positive integers n), the resulting a, b, c are simultaneously positive; this follows directly from the leading terms of the parametric expressions, which are positive for t > 0. We will add a short subsection (or paragraph) that isolates this infinite subfamily, states the sign conditions on t, and verifies positivity by direct inspection of the formulas. This supplies the explicit infinite subfamily requested and removes any ambiguity arising from sign patterns or denominators. revision: yes

Circularity Check

0 steps flagged

Explicit algebraic constructions are self-contained

full rationale

The paper derives infinite families of rational triples for k=4 via direct parametrizations (likely rational points on auxiliary varieties or polynomial identities) that are verified equation-by-equation to satisfy a b +1 = x^4 etc. No parameter is fitted to data and then relabeled a prediction; no uniqueness theorem is imported from the author's prior work to force the form; positivity is asserted by choosing parameters in open sets of the rationals where all three elements remain positive, which is an independent verification step rather than a tautology. The derivation chain therefore stands on its own algebraic content.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.0 · 5414 in / 955 out tokens · 46469 ms · 2026-05-10T06:38:16.784917+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Quartic Rational Diophantine Quadruples and the Euler Surface

    math.NT 2026-04 unverdicted novelty 8.0

    Infinitely many quartic rational Diophantine quadruples exist and are parametrized by rational points on the Euler surface X^4 + Y^4 = Z^4 + W^4.

Reference graph

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