Weak Hall's conjecture, approximation gains, and continued fractions

Abstract

For an integer solution of $y^2=x^3+k$ with $x,y\ge 1$ and $k\ne 0$, 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 $G_a=\log\max(x^3,y^2)/\log P$ 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 $G_a\le 1$. More generally, for $6/11\leκ<6/5$, an asymptotic bound $G_a\leκ$ gives $|k|>x^{3/κ-5/2-δ}$ for every $δ>0$ outside finitely many solutions, and for $3/4\leκ<6/5$ the two statements are equivalent. Since the approximation gain never exceeds the quality, an $abc$ inequality with exponent $1\leκ<6/5$ 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 $\sqrt x$.