AI 中文总结
本文研究abc猜想，通过定义根基过剩量、缺陷等概念分析梅森族三元组的越界情况，推导缺陷定律，指出abc反例族需无限多特定维弗里希素数，并区分两条相关椭圆曲线。
AI 中文摘要
对于互素的a+b=c，当a和b为奇数、c为偶数时，其奇偶类可进行精确处理。三元组由根基过剩量E_ε=log c - (1+ε)log rad(abc)度量，当固定ε>0时该值非负则称为“越界的（transgressive）”。abc猜想断言此类三元组的数量是有限的。过剩量可通过缺陷（根基所丢弃的重复素数的“质量”）精确表示，并重新整理为一个线性阈值，其中较小的加数s=min{a,b}明确出现。要么s在某个子序列中保持有界，要么缺陷必须超过阈值并以放大的指数重新满足阈值。对于梅森族V_m=(1,2^m-1,2^m)，当ε=0时越界恰好发生在2^m-1非无平方因子时，对应的指数集密度为47/210。缺陷服从精确定律Δ_m=Ω_m + log G_m，其中Ω_m是整除2^m-1的维弗里希素数的集合，G_m是m的最大除数，其素因子均整除2^m-1。该除数在素幂指数上是平凡的，在m_k=lcm(1,…,k)上最大，其中无维弗里希素数机制所能达到的余量q(V_{m_k}) - 1 ≥ (1-o(1))log m_k/(m_k log 2)。沿梅森线的abc反例族将需要无限多个具有指数阶缺陷增长的维弗里希素数。最后一节区分了该类对应的两条椭圆曲线：弗雷曲线（其最小判别式以2处的有界修正项表示总缺陷）和同余数雅可比（其斯皮罗商始终小于3）。
英文摘要
For coprime $a+b=c$, the parity class in which $a$ and $b$ are odd and $c$ is even admits an exact treatment. Triples are measured by the radical excess $E_{\varepsilon}=\log c-(1+\varepsilon)\log\operatorname{rad}(abc)$, and called transgressive at fixed $\varepsilon>0$ when it is non-negative. The $abc$ conjecture asserts that such triples are finite in number. The excess is written exactly in terms of the defect, the mass of repeated primes the radical discards, and rearranges into a linear threshold in which the smaller summand $s=\min\{a,b\}$ appears explicitly. Either $s$ stays bounded along a subsequence, or the defect must overshoot the threshold and rejoin it at an amplified exponent. For the Mersenne family $\mathcal{V}_m = (1, 2^m-1, 2^m)$ transgression at $\varepsilon=0$ holds precisely when $2^m-1$ fails to be squarefree, on a set of exponents of density $47/210$. The defect obeys the exact law $Δ_m = Ω_m + \log G_m$, where $Ω_m$ collects the Wieferich primes dividing $2^m-1$ and $G_m$ is the largest divisor of $m$ whose prime factors divide $2^m-1$. This divisor is trivial on prime-power exponents and largest on $m_k = \operatorname{lcm}(1,\dots,k)$, where the margin $q(\mathcal{V}_{m_k}) - 1 \geq (1-o(1))\log m_k / (m_k\log 2)$ is the most any Wieferich-free mechanism can give. An $abc$ counterexample family along the Mersenne line would require infinitely many Wieferich primes with exponential order--defect growth. A final section separates two elliptic curves attached to the class: the Frey curve, whose minimal discriminant expresses the total defect with a bounded correction at $2$, and a congruent-number Jacobian, whose Szpiro quotient stays below $3$.
