Hall's conjecture is an open question on the differences between perfect squares and perfect cubes, asserting that a perfect square and a perfect cube that are not equal must lie a substantial distance apart; it arose from the Mordell equation in the theory of integer points on elliptic curves. The original version, formulated by Marshall Hall Jr. in 1970, proposes a positive constant bounding the size of that gap, a bound later shown by Danilov in 1982 to have an optimal exponent.
Facts
StatementA perfect square y^2 and a perfect cube x^3 that are not equal must lie a substantial distance apart. 1 Proposed Year Progress Toward ResolutionThe original strong form has never been disproved but is no longer believed true; the weak form, with exponent below 1/2, is now generally meant and would follow from the ABC conjecture. 1 Difficulty Hint Classification
Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Source Hall's Conjecture (Wikipedia)
Sources
1. Hall's Conjecture (Wikipedia)
Wikimedia FoundationLead section
The original version of Hall's conjecture, formulated by Marshall Hall, Jr. in 1970
Lead section, first paragraph
must lie a substantial distance apart
Body, paragraph on the strong and weak forms
has never been disproved, although it is no longer believed to be true
Lead section, difficulty hint
an open question on the differences between perfect squares and perfect cubes
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