2025/12/11 by Valery Asiryan, Asiryan, Valery
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2512.11072
We study the Diophantine equation a5+b5=c5+d5 under the linear slicing constraint (c+d)-(a+b)=h. We first prove the necessary congruence 30| h. After symmetrization, the associated discriminant equation defines, for each fixed nonzero slice parameter h, a genus-one curve over ℚ(S); to study Mordell-Weil rank, one must pass to its Jacobian fibration Eh/ℚ(S). We show that Eh carries a global rational 2-torsion section and never has full rational 2-torsion over ℚ(S). We also prove that no nonsingular rational specialization acquires additional rational 2-torsion: by homogeneity, the relevant square condition reduces to rational points on a universal genus-two hyperelliptic curve, whose rational points are determined via a verified Magma computation using a rank-0 bound and the Chabauty-Coleman method. We further show that, after the normalization x=S/h, the Jacobian fibrations for all h≠ 0 become isomorphic over a rational function field. For the representative slice h=30, we compute the classical invariants of the associated binary quartic, obtain an explicit Weierstrass model, and apply the Gusić-Tadić injectivity criterion together with verified specialized-rank computations to prove the uniform bound rank Eh(ℚ(S))≤ 1 for all h≠ 0. We then construct an explicit rational section on the universal Jacobian model and, specializing at S=12 on the slice h=30, show via injective specialization that this section has infinite order. Consequently, rank Eh(ℚ(S))=1 for every h∈ℚ^×. We conclude by recording the additional integrality, parity, and size conditions required to recover integer solutions from the genus-one/Jacobian framework.