The quality plane of abc-triples and the decomposition into gains
Abstract
To a coprime triple a+b=c we attach two coordinates, the quality X and the logarithmic mass Y, whose ratio λ measures the balance of the triple and lies asymptotically between $2$ and $3$. In these coordinates the abc conjecture and Szpiro's conjecture for Frey curves become asymptotic bounds with thresholds X=1 and Y=3, and the direct implications between them follow from the range of λ. We then place the approximation and power gains of Müller, Taktikos and de Weger in the same plane. Relative to a chosen presentation of the triple, one exponent factor f relates the gains to the coordinates, and Szpiro's conjecture for this family is equivalent to the asymptotic bound f≤ 3 uniformly over all presentations. Under the two separate asymptotic gain bounds for a fixed choice of presentations, we obtain limsup X≤ 9/2 and limsup Y≤ 27/2 as the radical tends to infinity. Stronger conclusions depend on the presentation or on a joint bound for the gains. We close with numerical data and two research objectives.
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BibTeX
@misc{laniewski2026quality,
title = {{The quality plane of $abc$-triples and the decomposition into gains}},
author = {R. Laniewski and K. Müller},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03212},
eprint = {2610.03212},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03212},
}
RIS
TY - UNPB
TI - The quality plane of abc-triples and the decomposition into gains
AU - R. Laniewski
AU - K. Müller
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.03212
UR - https://arxiv.org/abs/2610.03212
AB - To a coprime triple a+b=c we attach two coordinates, the quality X and the logarithmic mass Y, whose ratio λ measures the balance of the triple and lies asymptotically between $2$ and $3$. In these coordinates the abc conjecture and Szpiro's conjecture for Frey curves become asymptotic bounds with thresholds X=1 and Y=3, and the direct implications between them follow from the range of λ. We then place the approximation and power gains of Müller, Taktikos and de Weger in the same plane. Relative to a chosen presentation of the triple, one exponent factor f relates the gains to the coordinates, and Szpiro's conjecture for this family is equivalent to the asymptotic bound f≤ 3 uniformly over all presentations. Under the two separate asymptotic gain bounds for a fixed choice of presentations, we obtain limsup X≤ 9/2 and limsup Y≤ 27/2 as the radical tends to infinity. Stronger conclusions depend on the presentation or on a joint bound for the gains. We close with numerical data and two research objectives.
ER -