Weak Hall's conjecture

Weak Hall’s conjecture asks for a uniform positive-power lower bound on the difference between a square and a cube. Marshall Hall formulated the original conjecture in 1970; the weaker form remains open and would follow from the $abc$ conjecture.

Known results

• Danilov (1982) constructed infinitely many examples with $0<|x^3-y^2|<0.97\sqrt{x}$.
• Stark proved a logarithmic lower bound, $|y^2-x^3|>C(\log x)^\kappa$ for every $\kappa<1$ (reported in 2010).
• Elkies (1998) found exceptional small differences; later computation found 24 examples with $x<10^{18}$.
• The standard weaker Hall form remains unresolved, despite these upper-bound constructions.

September 2026 developments

Laniewski and Müller prove an equivalence between Weak Hall’s conjecture and $G_a\le 1$, with broader approximation-gain bounds yielding explicit lower bounds on $|k|$; they claim no unconditional proof. A separate 2026 preprint obtains the unconditional estimate $|x^3-y^2|\gg (\log X)^2/(\log\log X)^8$, calling it a modest step toward Hall’s conjecture, not a solution.

Current status (as of September 2026): A logarithmic lower bound and new reformulations are available, but the required positive-power bound remains open.