Weak Hall's conjecture, approximation gains, and continued fractions
Abstract
For an integer solution of y2=x3+k with x,y≥ 1 and k≠ 0, the integers x3, |k| and y2 form a triple in which one term is the sum of the other two. When gcd(x,y)=1 this triple is coprime, and the abc conjecture predicts that its quality is asymptotically at most $1$. Following Müller, Taktikos and de Weger, we factor this quality through the product P=xy|k| into the approximation gain Ga=logmax(x3,y2)/log P and the power gain Gp=log P/lograd(xy|k|), which is at least $1$. We show that the weak form of Hall's conjecture, which asks that |k|>x1/2−δ for every δ>0 outside finitely many solutions, is equivalent to the asymptotic bound Ga≤ 1. More generally, for 6/11≤κ<6/5, an asymptotic bound Ga≤κ gives |k|>x3/κ−5/2−δ for every δ>0 outside finitely many solutions, and for 3/4≤κ<6/5 the two statements are equivalent. Since the approximation gain never exceeds the quality, an abc inequality with exponent 1≤κ<6/5 gives the same bound for primitive solutions. The case κ=1 extends to all solutions and recovers the classical implication from the abc conjecture to weak Hall. We also relate small values of |k| to large partial quotients of √x.
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BibTeX
@misc{laniewski2026weak,
title = {{Weak Hall's conjecture, approximation gains, and continued fractions}},
author = {R. Laniewski and K. Müller},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35252},
eprint = {2609.35252},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35252},
note = {CC0},
}
RIS
TY - UNPB
TI - Weak Hall's conjecture, approximation gains, and continued fractions
AU - R. Laniewski
AU - K. Müller
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35252
UR - https://arxiv.org/abs/2609.35252
AB - For an integer solution of y^2=x^3+k with x,y≥ 1 and k≠ 0, the integers x^3, |k| and y^2 form a triple in which one term is the sum of the other two. When gcd(x,y)=1 this triple is coprime, and the abc conjecture predicts that its quality is asymptotically at most $1$. Following Müller, Taktikos and de Weger, we factor this quality through the product P=xy|k| into the approximation gain G_a=logmax(x^3,y^2)/log P and the power gain G_p=log P/lograd(xy|k|), which is at least $1$. We show that the weak form of Hall's conjecture, which asks that |k|>x^1/2−δ for every δ>0 outside finitely many solutions, is equivalent to the asymptotic bound G_a≤ 1. More generally, for 6/11≤κ<6/5, an asymptotic bound G_a≤κ gives |k|>x^3/κ−5/2−δ for every δ>0 outside finitely many solutions, and for 3/4≤κ<6/5 the two statements are equivalent. Since the approximation gain never exceeds the quality, an abc inequality with exponent 1≤κ<6/5 gives the same bound for primitive solutions. The case κ=1 extends to all solutions and recovers the classical implication from the abc conjecture to weak Hall. We also relate small values of |k| to large partial quotients of √x.
ER -