Pairs of Pythagorean triangles with given catheti ratios
Pith reviewed 2026-05-25 11:07 UTC · model grok-4.3
The pith
For catheti ratios A/a and B/b with Aa ≠ Bb there exist infinitely many non-similar pairs of Pythagorean triangles satisfying the ratios.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
There are infinitely many essentially different (non-similar) pairs of Pythagorean triangles (a, b, c), (A, B, C) satisfying given proportions A/a, B/b, provided that Aa ≠ Bb.
What carries the argument
The Diophantine system formed by the two Pythagorean equations together with the two fixed-ratio equations, shown to possess infinitely many solutions once Aa ≠ Bb.
If this is right
- Any ratios meeting the product inequality produce infinitely many distinct pairs.
- The triangles in each pair can be taken primitive or with arbitrary common factors.
- Similarity is ruled out because the side ratios between the two triangles differ.
- The infinitude holds for both primitive and non-primitive triples.
Where Pith is reading between the lines
- When Aa = Bb the same system may possess only finitely many solutions.
- The method could be adapted to produce infinite families with additional constraints such as equal perimeters or equal areas.
- Analogous infinitude statements might hold for triples of triangles or for higher-dimensional Pythagorean objects.
Load-bearing premise
The chosen ratios A/a and B/b are such that the Diophantine system admits integer solutions at all.
What would settle it
An explicit pair of positive ratios A/a, B/b with Aa ≠ Bb for which the corresponding system of equations has only finitely many positive integer solutions a, b, c, A, B, C.
read the original abstract
In this note we investigate the problem of finding pairs of Pythagorean triangles $(a, b, c), (A, B, C)$, with given catheti ratios $A/a, B/b$. In particular, we prove that there are infinitely many essentially different ("non-similar") pairs of Pythagorean triangles $(a, b, c), (A, B, C)$ satisfying given proportions, provided that $Aa\neq Bb$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript claims to prove that, for given positive real catheti ratios A/a and B/b, there exist infinitely many essentially different (non-similar) pairs of Pythagorean triangles (a,b,c) and (A,B,C) satisfying those ratios, provided Aa ≠ Bb.
Significance. If the central claim held with a corrected non-similarity condition, the result would supply a Diophantine construction for infinite families of non-similar Pythagorean pairs with prescribed leg ratios, which is of modest interest in elementary number theory. The manuscript supplies no machine-checked proofs or parameter-free derivations.
major comments (1)
- [Abstract] Abstract (and main theorem statement): the proviso 'provided that Aa ≠ Bb' does not guarantee non-similarity. Similarity holds precisely when A/a = B/b (i.e., aB = Ab). When A/a = B/b but a ≠ b, the triangles are similar yet Aa ≠ Bb is satisfied, so the stated condition includes similar pairs and fails to support the claim of 'essentially different (non-similar)' pairs. This is load-bearing for the central result.
minor comments (1)
- The abstract supplies no explicit parametric formulas or verification steps for the claimed solutions; the full text should include at least one concrete numerical example satisfying both Pythagorean equations simultaneously.
Simulated Author's Rebuttal
We thank the referee for their careful reading of the manuscript and for identifying an important error in the statement of the non-similarity condition. We address the major comment below.
read point-by-point responses
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Referee: [Abstract] Abstract (and main theorem statement): the proviso 'provided that Aa ≠ Bb' does not guarantee non-similarity. Similarity holds precisely when A/a = B/b (i.e., aB = Ab). When A/a = B/b but a ≠ b, the triangles are similar yet Aa ≠ Bb is satisfied, so the stated condition includes similar pairs and fails to support the claim of 'essentially different (non-similar)' pairs. This is load-bearing for the central result.
Authors: We agree that the condition Aa ≠ Bb does not guarantee non-similarity of the triangles. As the referee correctly notes, the triangles are similar precisely when A/a = B/b (equivalently aB = Ab). When A/a = B/b but a ≠ b, the pairs satisfy Aa ≠ Bb yet remain similar, so the stated proviso fails to exclude similar pairs. We will revise the abstract and the main theorem statement to replace the condition 'provided that Aa ≠ Bb' with the correct condition 'provided that A/a ≠ B/b'. This correction is necessary and will be incorporated in the revised manuscript. The underlying Diophantine construction appears to produce pairs satisfying the corrected non-similarity condition, so the central existence result should hold after the change. revision: yes
Circularity Check
No circularity; derivation uses external Diophantine machinery
full rationale
The paper claims to prove existence of infinitely many non-similar Pythagorean triangle pairs with prescribed catheti ratios A/a and B/b, under the condition Aa ≠ Bb. No equations, parameters, or constructions in the provided abstract reduce the target result to a self-definition, a fitted input renamed as prediction, or a load-bearing self-citation chain. The proof is described as resting on standard number-theoretic tools for solving the associated Diophantine system, which are independent of the specific infinitude claim being established. The condition Aa ≠ Bb is an explicit hypothesis rather than a derived or fitted quantity, and the non-similarity claim is presented as following from that hypothesis via external results. This is the normal case of a self-contained existence proof.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Pythagorean triples satisfy a² + b² = c² with a, b, c positive integers
- domain assumption The catheti ratios A/a and B/b are fixed positive reals
discussion (0)
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