Conjecture 2.7 — Hall

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For positive integers $x,y$, compare the square $y^2$ and the cube $x^3$.

Statement recorded in the supplied source.

Is there an absolute constant $C>0$ such that every $x,y\ge1$ with $y^2\ne x^3$ satisfies \[|y^2-x^3|\ge C\max\{y^2,x^3\}^{1/6}?\] [S2]

Short English statement

Is every nonzero gap between a positive square and a positive cube at least a fixed constant times the sixth root of their larger value?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700015 (AMR-086-0015). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Conjecture 2.7. https://arxiv.org/abs/math/0312440.