Marshall Hall's conjecture
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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
- |m2 − n3| > C(n)√n
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
- |m2 − n3| > c(ε)n½ − ε.
The strong form, on which doubt has been cast, replaces the RHS with a constant multiple of √n. This latter formulation is the original one of Marshall Hall, Jr., from 1970.
A generalization to other perfect powers is Pillai's conjecture.
[edit] Reference
- Marshall Hall, Jr., The Diophantine equation x3 − y2=k In Computers in Number Theory (A.Atkin, B.Birch, eds.; Academic Press, 1971),
[edit] External links
- [1], page of Noam Elkies on the problem
- [2]
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