About the problem

Hall’s conjecture says that for any positive integers \(x\) and \(y\) such that \(y^2 \neq x^3\), the value of \(\frac{\sqrt{x}}{|y^2 - x^3|}\) is bounded above by an absolute constant; i.e. there exists a constant \(C\) such that for every \(x\) and \(y\),

This problem asks for a pair of positive integers \(x\) and \(y\) such that \(\frac{\sqrt{x}}{|y^2 - x^3|} > 100\), which would imply that if Hall’s conjecture were true, then \(C > 100\).

Currently, it is known that if Hall’s conjecture were true, then \(C > 46\); this lower bound on \(C\) was given by Noam Elkies in 1998, using an extremely large value of \(x\). Despite substantial effort, this bound has not been improved since then. For more information, see Noam Elkies’ page on Hall’s conjecture.

The verifier for this problem is an exact check; the primary risk is that Hall’s conjecture is true for some constant \(C \leq 100\), but this would be surprising. Another risk is that a solution gives no insight; the hope is that in order to beat the current record, some new idea or insight is needed.

Prompt

Hall's conjecture concerns how close a perfect square can be to a perfect cube without being equal to it. For positive integers x and y with

    x^3 != y^2,

define the "Hall ratio"

    r(x, y) = sqrt(x) / |y^2 - x^3|.

A large ratio means y^2 is a remarkably good (but not exact) approximation to x^3 relative to the size of x. The original strong form of Hall's conjecture asserted that this ratio is bounded by an absolute constant. Decades of computation have produced only finitely many large examples: the largest ratio currently known is approximately 46.6. A pair achieving a ratio greater than 100 would be a new record.

## Task

Find positive integers x and y such that

    x^3 != y^2    and    sqrt(x) / |y^2 - x^3| > 100.

## Submission

Submit your two integers as (x, y) using the submit tool. There is no bound on the magnitude of x or y. Your submission is checked exactly with arbitrary-precision integer arithmetic (the ratio condition is tested as x > (100 * |y^2 - x^3|)^2), so an approximate or rounded answer will not be accepted.