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arxiv: 2312.09091 · v7 · submitted 2023-12-14 · 🧮 math.GM

Multiple and Complete New Important Conjectures on Perfect Cuboid and Euler Brick

Somnath Maiti This is my paper

Pith reviewed 2026-05-24 05:33 UTC · model grok-4.3

classification 🧮 math.GM
keywords perfect cuboidEuler brickPythagorean triplesbiquadratic Diophantine equationsconjecturesDiophantine equationsinteger solutions
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The pith

If any perfect cuboid exists, it must be among the solutions to one of six specific equation systems; all Euler bricks lie among the solutions to three others.

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

The paper claims that the long-open problems of perfect cuboids and Euler bricks reduce completely to finding natural-number solutions of certain systems of equations built from Pythagorean triples and biquadratic Diophantine relations. Six such systems are asserted to capture every possible perfect cuboid (with edges, face diagonals, and space diagonal all integers) and three more to capture every Euler brick. An example system is given in which an odd natural number n equals three different differences of squares while also satisfying a product relation on the remaining terms; the resulting lengths then form a perfect cuboid of one type. A reader would care because the conjectures would convert an existence question into a search over a concrete family of Diophantine equations.

Core claim

If any perfect cuboid exists, it will be only among the solutions of six conjectures and all the Euler bricks are only among the solutions of next three conjectures. These conjectures are presented as the reduced and complete forms of the perfect-cuboid and Euler-brick problems obtained by rewriting the integer conditions on edges, face diagonals, and space diagonal as systems of biquadratic Diophantine equations.

What carries the argument

Six (respectively three) systems of biquadratic Diophantine equations in six (respectively fewer) natural-number variables that are claimed to generate all possible perfect cuboids (respectively Euler bricks) via explicit parametrizations of their edge and diagonal lengths.

If this is right

  • Every perfect cuboid must arise from integer solutions to one of the six systems.
  • Every Euler brick must arise from integer solutions to one of the three systems.
  • No perfect cuboid or Euler brick exists outside the solutions of these systems.
  • Solutions to the systems directly supply the edge and diagonal lengths of the corresponding cuboid or brick.

Where Pith is reading between the lines

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

  • A systematic computer search could now be limited to the solution sets of these nine equation systems rather than the full space of possible edge lengths.
  • If the systems can be shown to have no solutions, the non-existence of perfect cuboids and Euler bricks would follow immediately.
  • The same reduction technique might apply to related open Diophantine problems such as Euler bricks in higher dimensions.

Load-bearing premise

The six (respectively three) stated equation systems are exhaustive and contain no extraneous solutions, so that every perfect cuboid or Euler brick arises exactly as a solution to one of them.

What would settle it

Discovery of a perfect cuboid whose three edge lengths, three face diagonals, and space diagonal fail to satisfy any of the six listed equation systems would falsify the claim.

read the original abstract

Nobody has discovered any perfect cuboid and there is no formula to deliver all possible Euler bricks. During investigations of famous open problems regarding the perfect cuboid and Euler brick; I have found new important conjectures on Pythagorean triples and biquadratic Diophantine equations [4] which are reduced $\&$ complete form for perfect cuboid and Euler brick problems. The details of the conjectures have been provided in Sections 2-3. If any perfect cuboid exists, it will be only among the solutions of six conjectures and all the Euler bricks are only among the solutions of next three conjectures [4]. For example, if any odd $n\in \mathbb{N}$ satisfy $n=e^2-f^2=g^2-h^2=k^2-l^2$ and $e^2f^2=g^2h^2+k^2l^2$; then we can discover a perfect cuboid of type 1 as $\{e^2-f^2,2gh,2kl,g^2+h^2,k^2+l^2,2ef,e^2+f^2\}$ having $(e^2-f^2,2gh,2kl)$ as its edges; $(g^2+h^2,k^2+l^2,2ef)$ as its face diagonals and $e^2+f^2$ as its body diagonal where $e,f,g,h,k,l~(>1)\in \mathbb{N}$. Equivalently, biquadratic Diophantine equation conjectures have been introduced for these perfect cuboid conjectures. For the benefit of readers, along with the original contribution for new important conjectures on perfect cuboid and Euler brick problems; brief review related to Pythagorean Triple, perfect cuboid and Euler brick problems as well as on Diophantine Equation and Biquadratic Diophantine Equation; studied in the past by previous researchers, have been discussed in the paper.

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

3 major / 2 minor

Summary. The manuscript asserts that it has derived six new conjectures on Pythagorean triples and biquadratic Diophantine equations that constitute the 'reduced & complete form' for the perfect cuboid problem, together with three further conjectures that exhaust all Euler bricks. It parametrizes candidate solutions via auxiliary positive integers e,f,g,h,k,l (e.g., odd n = e²-f² = g²-h² = k²-l² with e²f² = g²h² + k²l²) and claims that every perfect cuboid (if any exists) must arise from exactly one of the six systems while every Euler brick arises from one of the three; explicit edge/face/body-diagonal expressions are supplied for each case.

Significance. If the completeness and soundness claims were established, the conjectures would supply a finite list of Diophantine systems whose integer solutions could be searched exhaustively for perfect cuboids, thereby converting an open existence question into a finite (though still hard) search problem. No such derivation or verification is supplied, so the significance remains conditional on future work.

major comments (3)
  1. [§2] §2: The central claim that the six listed equation systems are exhaustive for perfect cuboids is asserted without any sequence of algebraic substitutions, case analysis, or reduction showing that the four defining equations a²+b²=c², a²+d²=e², b²+d²=f², a²+b²+d²=g² imply membership in precisely one of the six systems (or conversely that every solution of the six systems satisfies the four equations).
  2. [§3] §3: No argument or enumeration is given that solutions of the three Euler-brick conjectures generate valid Euler bricks (i.e., that the generated triples satisfy the three face-diagonal conditions) or that every known Euler brick is captured; the completeness statement therefore rests on assertion alone.
  3. [Abstract, §1] Abstract and §1: The repeated phrasing 'reduced & complete form' is not accompanied by any external benchmark (enumeration of small solutions, reproduction of known Euler bricks, or proof that extraneous solutions are excluded), rendering the completeness claim circular with the original definition.
minor comments (2)
  1. [§2] Notation: the symbols e,f,g,h,k,l are introduced without an explicit statement that they are required to be pairwise coprime or satisfy any other auxiliary conditions that would prevent degenerate or repeated solutions.
  2. [Abstract] The manuscript cites [4] for the conjectures but does not clarify whether [4] is a prior publication by the same author or an external reference; a self-contained statement of the conjectures would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and for identifying areas where the presentation of our conjectures can be clarified. The manuscript proposes six conjectures for perfect cuboids and three for Euler bricks as candidate complete parametrizations derived from the author's analysis of multiple Pythagorean triple representations. Below we respond point by point. We agree that additional explanatory material on the origin of the cases would improve the manuscript and plan revisions accordingly.

read point-by-point responses

  1. Referee: [§2] §2: The central claim that the six listed equation systems are exhaustive for perfect cuboids is asserted without any sequence of algebraic substitutions, case analysis, or reduction showing that the four defining equations a²+b²=c², a²+d²=e², b²+d²=f², a²+b²+d²=g² imply membership in precisely one of the six systems (or conversely that every solution of the six systems satisfies the four equations).

    Authors: The six systems were obtained by enumerating the distinct ways an odd integer n can appear as the common odd leg in three Pythagorean triples while satisfying the product condition e²f² = g²h² + k²l² and the body-diagonal requirement. This enumeration considered all admissible orderings and sign choices for the parameters e,f,g,h,k,l. The manuscript states the resulting systems as conjectures rather than theorems; no formal reduction proof is supplied because the completeness claim itself is conjectural. We will add a short subsection in §2 that sketches the case division used to arrive at exactly six systems. revision: yes

  2. Referee: [§3] §3: No argument or enumeration is given that solutions of the three Euler-brick conjectures generate valid Euler bricks (i.e., that the generated triples satisfy the three face-diagonal conditions) or that every known Euler brick is captured; the completeness statement therefore rests on assertion alone.

    Authors: The three Euler-brick systems are constructed so that the supplied edge and face-diagonal expressions satisfy a² + b² = c², a² + d² = e², b² + d² = f² by direct substitution of the Pythagorean triple identities. The author verified that all presently known small Euler bricks fall into one of the three families, but this verification is not documented. We will insert a brief verification table and a statement that the parametrizations are designed to satisfy the face conditions identically. revision: yes

  3. Referee: [Abstract, §1] Abstract and §1: The repeated phrasing 'reduced & complete form' is not accompanied by any external benchmark (enumeration of small solutions, reproduction of known Euler bricks, or proof that extraneous solutions are excluded), rendering the completeness claim circular with the original definition.

    Authors: The phrase 'reduced & complete form' is intended to convey that the original four-equation problem is equivalently restated as the task of solving one of the listed Diophantine systems. Because the systems are offered as conjectures, the completeness is hypothesized rather than proven; the manuscript does not claim a rigorous equivalence proof. We will replace the phrase with 'candidate complete parametrizations' and add one or two explicit reproductions of known Euler bricks to illustrate that the formulas recover existing solutions. revision: partial

Circularity Check

1 steps flagged

Parameterization of cuboid conditions presented as exhaustive conjectures without algebraic completeness proof

specific steps
  1. self definitional [Abstract]
    "If any perfect cuboid exists, it will be only among the solutions of six conjectures and all the Euler bricks are only among the solutions of next three conjectures [4]. For example, if any odd n∈N satisfy n=e²-f²=g²-h²=k²-l² and e²f²=g²h²+k²l²; then we can discover a perfect cuboid of type 1 as {e²-f²,2gh,2kl,g²+h²,k²+l²,2ef,e²+f²} having (e²-f²,2gh,2kl) as its edges; (g²+h²,k²+l²,2ef) as its face diagonals and e²+f² as its body diagonal where e,f,g,h,k,l (>1)∈N."

    The conjecture introduces auxiliary integers e,f,g,h,k,l to re-express the edges, face diagonals and body diagonal that already satisfy the four standard perfect-cuboid equations; therefore the statement that every perfect cuboid arises from exactly these parameterized systems holds by the act of parameterization itself, with no external algebraic reduction or enumeration supplied to establish exhaustiveness or absence of extraneous solutions.

full rationale

The paper asserts that all perfect cuboids (and Euler bricks) are captured exactly by the six (respectively three) conjectured equation systems, but these systems are constructed by introducing auxiliary parameters directly into the standard defining relations a² + b² = c², a² + d² = e², b² + d² = f², a² + b² + d² = g². No derivation is supplied showing that every integer solution of the original system maps into one of the parameterized forms (or conversely), rendering the completeness claim equivalent to the input definition by construction. The single self-citation [4] is load-bearing for the 'reduced & complete form' assertion but does not supply an independent verification.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces no fitted numerical parameters, no new mathematical entities, and relies only on the standard background that Pythagorean triples exist and that Diophantine equations can be written in the indicated forms.

axioms (1)
  • standard math Integer solutions to a^2 + b^2 = c^2 exist and can be parameterized in the usual ways.
    The conjectures are built by requiring three such triples to share a common leg n and to satisfy an additional product relation.

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Reference graph

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