Marshall Hall's conjecture

From Wikipedia, the free encyclopedia

In mathematics, Marshall Hall's conjecture is an open question, as of 2006, on the differences between perfect squares and perfect cubes. Aside from the case of a perfect sixth power, it asserts that a perfect square $m 2$ and a perfect cube $n 3$ 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 $n\rightarrow\infty$ . That is, for any given $ε > 0$ , we can assert that

c (ε) n 1 / 2 - ε < | 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.

A generalization to other perfect powers is Pillai's conjecture.

[edit] References

Hall, Jr., Marshall (1971). "The Diophantine equation $x 3 - y 2 = k$ ", in A. O. L. Atkin, B. J. Birch: Computers in Number Theory. ISBN 0120657503.