Marshall Hall's conjecture

In mathematics, Marshall Hall's conjecture is an open question, as of 2006^[update], on the differences between perfect squares and perfect cubes. Aside from the case of a perfect sixth power, it asserts that a perfect square m² and a perfect cube n³ must lie a substantial distance apart. This question arose from consideration of the Mordell equation in the theory of integer points on elliptic curves.

The weak form of Hall's conjecture is formulated as

$C(n)\sqrt{n} < |m^2 - n^3|$

where C(n) is an exponential factor less than 1, but which tends to 1 as n → ∞. That is, for any given ε > 0, we can assert that

$c(\varepsilon) n^{1/2-\varepsilon} < |m^2 - n^3|.\,$

The strong form, on which doubt has been cast, replaces the LHS with a constant multiple of $\sqrt{n}$ . This was the original formulation of Marshall Hall, Jr. in 1970.

The weak form of the conjecture would follow from the ABC conjecture. A generalization to other perfect powers is Pillai's conjecture.

References

Hall, Jr., Marshall (1971). "The Diophantine equation x³ - y² = k". In A. O. L. Atkin, B. J. Birch. Computers in Number Theory. ISBN 0120657503.

External links

[1], page of Noam Elkies on the problem.
[2], table of good examples of Marshall Hall's conjecture by Ismael Jimenez Calvo.