Abstract:The solvability of Diophantine quartic equations is a contemporary area of interest due to its connection with generalized Fermat's equation. In this work, we are interested in the integer solutions of a similar Diophantine equation p u^2 = v^2 + w^2. For a particular form of u, v, and w, we prove that the elliptic curves E_p: y^2 = x(x-1)(x+p^2), which arise from Heron triangles, for primes p = 1 (mod 8) where q = (p^2+1)/2 is also prime, exhibit a sharp dichotomy based on the solution of the aforementioned Diophantine equation: either rank(E_p(Q)) = 2 with trivial Shafarevich-Tate group or rank = 0 with III(E_p/Q)[2] = (Z/2Z)^2.
From: Vinodkumar Ghale [view email]
[v1]
Mon, 9 Jan 2023 16:19:41 UTC (11 KB)
[v2]
Tue, 10 Jan 2023 17:05:15 UTC (10 KB)
[v3]
Fri, 16 Jan 2026 17:33:59 UTC (13 KB)
[v4]
Tue, 31 Mar 2026 09:17:40 UTC (13 KB)
[v5]
Sun, 24 May 2026 12:28:10 UTC (10 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。