Marshall Hall's conjecture

Marshall Hall's conjecture

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 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.
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