Radical defects, Wieferich primes, and the $abc$ conjecture

Abstract

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 $E_{\epsilon }=\log c-(1+\epsilon )\log \mathrm{rad}(abc)$, and called transgressive at fixed $\epsilon >0$ 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 $\epsilon =0$ 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 ${\Delta }_m={\Omega }_m+\log G_m$, where ${\Omega }_m$ 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 $m_k=\mathrm{lcm}(1,\dots ,k)$, where the margin $q({\mathcal{V}}_{m_k})-1\ge (1-o(1))\log m_k/(m_k\log 2)$ 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 $3$.