Exponent-one blockers and a Mordell-Weil construction of Euler bricks
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| Format: | Preprint |
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2026
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| _version_ | 1866910183637123072 |
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| author | Peschmann, René |
| author_facet | Peschmann, René |
| contents | A body cuboid is a rectangular parallelepiped with integer edges and integer face diagonals; if its space diagonal is also integer, it is a perfect cuboid, whose existence is a long-standing open problem. We make two contributions to the study of body cuboids parametrised by two coprime Pythagorean pairs $(a,b)$ and $(m,n)$ in Euclid form (Master-Hits).
The first is a verified exponent-one blocker phenomenon: for every Master-Hit, the space-diagonal norm $f_1 := (W_1 U_2)^2 + (U_1 V_2)^2$ admits a prime divisor $\ell$ of exponent exactly one which is coprime to a fixed list of $29$ canonical expressions in the parameters. This is strictly stronger than the existence of any odd-exponent prime divisor: a prime of exponent $3, 5, \ldots$ would obstruct $f_1$ from being a square but carry an extra square factor; the observed obstruction is always primitive. The phenomenon is verified on all $151{,}575$ Master-Hits whose $f_1$ has been fully factorised. Two natural strengthenings fail: the largest outside-parameter prime need not be a blocker, and the smallest outside-parameter blocker need not have exponent one.
The second contribution uses the elliptic fibration of the Master-Hit variety over the $(m,n)$-plane. For coprime $(m,n)$ the Master-Hit equation defines a genus-one quartic $H_{m,n}$; a quartic-to-Weierstrass normalisation gives an elliptic model $E_{m,n}$ with a rational function $τ$ returning $t^2$. Our generator enumerates bounded Mordell-Weil combinations on $E_{m,n}(\mathbb{Q})$, lifts the points satisfying $τ(P) \in \mathbb{Q}_{>0}^{\square}$ to admissible Euclid pairs $(a,b)$, and certifies each via exact integer arithmetic. From $61{,}829$ classical Master-Hits we generate $1{,}222{,}841$ further ones über $411$ fibres. None of the resulting $1{,}284{,}670$ Master-Hits is a perfect cuboid; all fully factored records satisfy the exponent-one blocker phenomenon. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_00573 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Exponent-one blockers and a Mordell-Weil construction of Euler bricks Peschmann, René Number Theory 11G05 (Primary), 11D72, 11Y50, 14H45 (Secondary) A body cuboid is a rectangular parallelepiped with integer edges and integer face diagonals; if its space diagonal is also integer, it is a perfect cuboid, whose existence is a long-standing open problem. We make two contributions to the study of body cuboids parametrised by two coprime Pythagorean pairs $(a,b)$ and $(m,n)$ in Euclid form (Master-Hits). The first is a verified exponent-one blocker phenomenon: for every Master-Hit, the space-diagonal norm $f_1 := (W_1 U_2)^2 + (U_1 V_2)^2$ admits a prime divisor $\ell$ of exponent exactly one which is coprime to a fixed list of $29$ canonical expressions in the parameters. This is strictly stronger than the existence of any odd-exponent prime divisor: a prime of exponent $3, 5, \ldots$ would obstruct $f_1$ from being a square but carry an extra square factor; the observed obstruction is always primitive. The phenomenon is verified on all $151{,}575$ Master-Hits whose $f_1$ has been fully factorised. Two natural strengthenings fail: the largest outside-parameter prime need not be a blocker, and the smallest outside-parameter blocker need not have exponent one. The second contribution uses the elliptic fibration of the Master-Hit variety over the $(m,n)$-plane. For coprime $(m,n)$ the Master-Hit equation defines a genus-one quartic $H_{m,n}$; a quartic-to-Weierstrass normalisation gives an elliptic model $E_{m,n}$ with a rational function $τ$ returning $t^2$. Our generator enumerates bounded Mordell-Weil combinations on $E_{m,n}(\mathbb{Q})$, lifts the points satisfying $τ(P) \in \mathbb{Q}_{>0}^{\square}$ to admissible Euclid pairs $(a,b)$, and certifies each via exact integer arithmetic. From $61{,}829$ classical Master-Hits we generate $1{,}222{,}841$ further ones über $411$ fibres. None of the resulting $1{,}284{,}670$ Master-Hits is a perfect cuboid; all fully factored records satisfy the exponent-one blocker phenomenon. |
| title | Exponent-one blockers and a Mordell-Weil construction of Euler bricks |
| topic | Number Theory 11G05 (Primary), 11D72, 11Y50, 14H45 (Secondary) |
| url | https://arxiv.org/abs/2605.00573 |