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 Eε=log c−(1+ε)lograd(abc), and called transgressive at fixed ε>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 𝒱m = (1, 2m−1, 2m) transgression at ε=0 holds precisely when 2m−1 fails to be squarefree, on a set of exponents of density 47/210. The defect obeys the exact law Δm = Ωm + log Gm, where Ωm collects the Wieferich primes dividing 2m−1 and Gm is the largest divisor of m whose prime factors divide 2m−1. This divisor is trivial on prime-power exponents and largest on mk = lcm(1,…,k), where the margin q(𝒱mk) − 1 ≥ (1−o(1))log mk / (mklog 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 $3$.
Related
- Perfect numbers, Wieferich primes and the solutions of binom2nn≡ 2n bmod n
- Generalized Wieferich primes and monogenic polynomials
- On Greenberg's conjecture for rational elliptic curves at Eisenstein primes
- The Poisson Tail Conjecture for primes in short intervals
- Kolyvagin's conjecture at non-ordinary primes
BibTeX
@misc{laniewski2026radical,
title = {{Radical defects, Wieferich primes, and the $abc$ conjecture}},
author = {R. Laniewski},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00039},
eprint = {2609.00039},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00039},
}
RIS
TY - UNPB
TI - Radical defects, Wieferich primes, and the abc conjecture
AU - R. Laniewski
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00039
UR - https://arxiv.org/abs/2609.00039
AB - 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_ε=log c−(1+ε)lograd(abc), and called transgressive at fixed ε>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 𝒱_m = (1, 2^m−1, 2^m) transgression at ε=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 = lcm(1,…,k), where the margin q(𝒱_m_k) − 1 ≥ (1−o(1))log m_k / (m_klog 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 $3$.
ER -