AI 中文总结
该研究指出Paulsen和West关于完美长方体对应椭圆曲线秩至少为2的证明存在漏洞，通过反例和完整论证证明了正长方体构型对应椭圆曲线的子群秩至少为2。
AI 中文摘要
Paulsen和West将完美长方体与同余数椭圆曲线上的有理点三元组关联，并断言对应曲线的秩至少为2。他们的定理4.2的证明存在一处漏洞：将生成元平移一个二阶点仅会交换两个坐标序列的奇数索引项，而非整个序列。我们针对该中间断言给出两个明确的反例，并完整证明了秩障碍。除子多项式论证将假设的秩1构型简化为剩余的一种奇偶模式。Ingram的本原除子定理随后迫使这些索引间产生加法关系，这与椭圆曲线上的正性引理相矛盾。更准确地说，每个正长方体构型的三个点生成的子群秩至少为2，即使其所在曲线具有更大的秩，该证明是无条件的。
英文摘要
Paulsen and West associate perfect cuboids with triples of rational points on a congruent-number elliptic curve and assert that the corresponding curve has rank at least two. The proof of their Theorem 4.2 contains a gap: translation of a generator by a point of order two interchanges only the odd-indexed terms of two coordinate sequences, rather than the entire sequences. We exhibit two explicit counterexamples to that intermediate assertion and give a complete proof of the rank obstruction. The division-polynomial argument reduces a hypothetical rank-one configuration to one remaining parity pattern. Ingram's primitive-divisor theorem then forces an additive relation among its indices, which is incompatible with a positivity lemma on the elliptic curve. More precisely, the three points of every positive cuboid configuration generate a subgroup of rank at least two, even when the ambient curve has larger rank. The proof is unconditional.
Comments12 pages
