arXiv subjects

René Peschmann

Publications and source records attributed to René Peschmann.

3 recordsLinked to original sources

arXiv2026

Exponent-one blockers and a Mordell-Weil construction of Euler bricks

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.

math.NT↗

arXiv2026

A torsion-intersection proof of perfect-cuboid nonexistence on 1,072 explicit master-tuple fibers

Building on the genus-3 reduction $C_A : w^2 = λ^8 + A λ^4 + 1$ established in our companion paper (arXiv:2604.09328), we give an unconditional proof of the perfect-cuboid conjecture ("Conjecture B") on $1{,}072$ explicit master-tuple fibers, excluding all rational $(a,b)$-specialisations on each such fiber. Our three main contributions are: (i) a structural classification theorem showing that every primitive Euler-brick arises from the standard $(a,b,m,n)$-parametrisation up to scaling; (ii) a torsion-intersection argument applied to the elliptic quotients $E_A'$ and $E_A''$: whenever the rank-zero hypothesis and the appropriate torsion condition hold for one of them, $|H_{m,n}(\mathbb{Q})| = 8$ is forced, with the eight points all corresponding to degenerate bricks; (iii) two complementary techniques to verify the rank-zero hypothesis algorithmically -- PARI's ellrank (2-descent) and, where this is ambiguous, Sage's exact rational evaluation of $L(E,1)/Ω_E$ via modular symbols, which combined with the modularity theorem, Kolyvagin's theorem, and Edixhoven's bound on the Manin constant for semistable curves yields an unconditional rank-zero certificate -- together with an explicit lift count refining the naive torsion-intersection bound when the torsion is larger than the leading case. We exhibit $1{,}072$ such fibers with $\max(m,n) \le 100$ on which Conjecture B is thereby established unconditionally.

math.NT↗

arXiv2026

Quartic reductions and elliptic obstructions for perfect Euler bricks

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.

math.NT↗