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| In [[mathematics]], '''Hall's conjecture''' is an open question, {{As of|2012|lc=on}}, on the differences between [[Square number|perfect squares]] and [[perfect cube]]s. It asserts that a perfect square ''y''<sup>2</sup> and a perfect cube ''x''<sup>3</sup> that are not equal must lie a substantial distance apart. This question arose from consideration of the [[Mordell equation]] in the theory of [[integer point]]s on [[elliptic curve]]s.
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| The original version of Hall's conjecture, formulated by [[Marshall Hall, Jr.]] in 1970, says that there is a positive constant ''C'' such that for any integers ''x'' and ''y'' for which ''y''<sup>2</sup> ≠ ''x''<sup>3</sup>,
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| :<math> |y^2 - x^3| > C\sqrt{|x|}.</math>
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| Hall suggested that perhaps ''C'' could be taken as 1/5, which was consistent with all the data known at the time the conjecture was proposed. Danilov showed in 1982 that the exponent 1/2 on the right side (that is, the use of |''x''|<sup>1/2</sup>) can't be replaced by any higher power: for no δ > 0 is there a constant ''C'' such that |''y''<sup>2</sup> - ''x''<sup>3</sup>| > C|''x''|<sup>1/2 + δ</sup> whenever ''y''<sup>2</sup> ≠ ''x''<sup>3</sup>.
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| In 1965, Davenport proved an analogue of the above conjecture in the case of polynomials:
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| if ''f''(''t'') and ''g''(''t'') are nonzero polynomials over '''C''' such that
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| ''g''(''t'')<sup>3</sup> ≠ ''f''(''t'')<sup>2</sup> in '''C'''[''t''], then
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| :<math> \deg(g(t)^2 - f(t)^3) \geq \frac{1}{2}\deg f(t) + 1.</math>
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| The ''weak'' form of Hall's conjecture, due to Stark and Trotter around 1980, replaces the square root on the right side of the inequality by any exponent ''less'' than 1/2: for any ''ε'' > 0, there is some constant ''c''(ε) depending on ε such that for any integers ''x'' and ''y'' for which ''y''<sup>2</sup> ≠ ''x''<sup>3</sup>,
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| :<math> |y^2 - x^3| > c(\varepsilon) x^{1/2-\varepsilon}.</math>
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| The original, ''strong'', form of the conjecture with exponent 1/2 has never been disproved, although it is no longer believed to be true and the term ''Hall's conjecture'' now generally means the version with the ε in it. For example, in 1998 Elkies found the example
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| 447884928428402042307918<sup>2</sup> - 5853886516781223<sup>3</sup> = -1641843,
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| for which compatibility with Hall's conjecture would require ''C'' to be less than .0214 ≈ 1/50, so roughly 10 times smaller than the original choice of 1/5 that Hall suggested.
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| The weak form of Hall's conjecture would follow from the [[ABC conjecture]].<ref>{{cite book | last=Schmidt | first=Wolfgang M. | authorlink=Wolfgang M. Schmidt | title=Diophantine approximations and Diophantine equations | series=Lecture Notes in Mathematics | volume=1467 | publisher=[[Springer-Verlag]] | year=1996 | edition=2nd | isbn=3-540-54058-X | zbl=0754.11020 | pages=205–206 }}</ref> A generalization to other perfect powers is [[Pillai's conjecture]].
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{cite book |last=Guy | first=Richard K. | authorlink=Richard K. Guy | title=Unsolved problems in number theory | publisher=[[Springer-Verlag]] |edition=3rd | year=2004 |isbn=978-0-387-20860-2 | zbl=1058.11001 | at=D9 }}
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| * {{cite book| last=Hall, Jr. | first=Marshall | year=1971 | chapter=The Diophantine equation ''x''<sup>3</sup> - ''y''<sup>2</sup> = ''k'' | pages=173–198 | title=Computers in Number Theory | editor1-first=A.O.L. | editor1-last=Atkin | editor1-link=A. O. L. Atkin | editor2-first=B. J. |editor2-last=Birch | editor2-link=Bryan John Birch | isbn=0-12-065750-3 | zbl=0225.10012}}
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| ==External links==
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| * [http://www.math.harvard.edu/~elkies/hall.html], page of [[Noam Elkies]] on the problem.
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| * [http://ijcalvo.galeon.com/hall.htm], table of ''good examples'' of ''Marshall Hall's conjecture'' by Ismael Jimenez Calvo.
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| [[Category:Number theory]]
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| [[Category:Conjectures]]
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