Quartic reductions and elliptic obstructions for perfect Euler bricks
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910119318519808 |
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| author | Peschmann, René |
| author_facet | Peschmann, René |
| contents | We show that the perfect Euler brick (perfect cuboid) problem is equivalent to the following elementary question: do there exist coprime integers $a, b, m, n$ such that the two expressions $(2(a^2-b^2)mn)^2 + ((a^2+b^2)(m^2-n^2))^2$ and $(4abmn)^2 + ((a^2+b^2)(m^2-n^2))^2$ are simultaneously perfect squares? Despite their near-identical structure (differing only in the first summand), no solution has ever been found. We reduce this quartic pair to a one-parameter family of genus-3 hyperelliptic curves $C_A\colon w^2 = λ^8 + Aλ^4 + 1$ and develop obstructions on the distinguished elliptic quotient $E_A$: the Kummer character $χ_f$ is non-trivial on the 4-torsion, and 2-descent arguments exclude several families of square classes. Computationally, we verify that no solution exists for parameters up to $10^3$. These results do not yet exclude perfect Euler bricks unconditionally; the remaining gap and possible approaches (including a genus-5 covering obstruction and connections to $\mathbb{Q}(\sqrt{2})$) are discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_09328 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quartic reductions and elliptic obstructions for perfect Euler bricks Peschmann, René Number Theory 11D09 (Primary), 11G05, 11G30 (Secondary) We show that the perfect Euler brick (perfect cuboid) problem is equivalent to the following elementary question: do there exist coprime integers $a, b, m, n$ such that the two expressions $(2(a^2-b^2)mn)^2 + ((a^2+b^2)(m^2-n^2))^2$ and $(4abmn)^2 + ((a^2+b^2)(m^2-n^2))^2$ are simultaneously perfect squares? Despite their near-identical structure (differing only in the first summand), no solution has ever been found. We reduce this quartic pair to a one-parameter family of genus-3 hyperelliptic curves $C_A\colon w^2 = λ^8 + Aλ^4 + 1$ and develop obstructions on the distinguished elliptic quotient $E_A$: the Kummer character $χ_f$ is non-trivial on the 4-torsion, and 2-descent arguments exclude several families of square classes. Computationally, we verify that no solution exists for parameters up to $10^3$. These results do not yet exclude perfect Euler bricks unconditionally; the remaining gap and possible approaches (including a genus-5 covering obstruction and connections to $\mathbb{Q}(\sqrt{2})$) are discussed. |
| title | Quartic reductions and elliptic obstructions for perfect Euler bricks |
| topic | Number Theory 11D09 (Primary), 11G05, 11G30 (Secondary) |
| url | https://arxiv.org/abs/2604.09328 |