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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~~.		[http://Www.Allstate.com/tools-and-resources/car-insurance/whats-in-a-new-car-warranty.aspx Friends contact] him Royal Cummins. Interviewing is what she does in her day job but soon her spouse and extended car warranty her will start their own business. Her buddies say it's not great for her but what she loves performing is flower arranging and she is trying to make it a occupation. Her family life in Delaware but she needs to [http://www.carsprotectionplus.com/ transfer] because of her family.<br><br>Feel free to [http://www.newsfortrader.com/ActivityFeed/MyProfile/tabid/60/userId/29712/language/it-IT/Default.aspx extended auto warranty] car warranty extended car [http://brunswickjets.com/UserProfile/tabid/1126/userId/227159/Default.aspx auto warranty] surf to my weblog ... [http://14.63.168.193/index.php?document_srl=249897&mid=freeboard 168.193]

	~~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>,~~

	~~:<math> \|y^2~~ - ~~x^3\| > C\sqrt{\|x\|}.</math>~~

	~~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>~~.

	~~In 1965, Davenport proved an analogue of the above conjecture~~ in ~~the case of polynomials:~~
	~~if ''f''(''t'')~~ and ~~''g''(''t'') are nonzero polynomials over '''C''' such that~~
	~~''g''(''t'')<sup>3</sup> ≠ ''f''(''t'')<sup>2</sup> in '''C'''[''t''], then~~

	~~:<math> \deg(g(t)^2 - f(t)^3) \geq \frac{1}{2}\deg f(t) + 1~~.~~</math>~~

	~~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>,~~

	~~:<math> \|y^2 - x^3\| > c(\varepsilon) x^{1/2-\varepsilon}.</math>~~

	~~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~~

	~~447884928428402042307918<sup>2<~~/~~sup> - 5853886516781223<sup>3<~~/~~sup> = -1641843,~~

	~~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.~~

	~~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]]~~.

	~~==Notes==~~
	~~{{reflist}}~~

	~~==References==~~
	* {{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 }}~~
	* {{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}}~~

	~~==External links==~~
	* [http://~~www~~.~~math.harvard.edu~~/~~~elkies~~/~~hall~~.~~html~~]~~, page of [[Noam Elkies]] on the problem~~.
	* [http://~~ijcalvo~~.~~galeon~~.~~com~~/~~hall~~.~~htm], table of ''good examples'' of ''Marshall Hall's conjecture'' by Ismael Jimenez Calvo~~.

	~~[[Category:Number theory]]~~
	~~[[Category:Conjectures]~~]

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2013-12-10T22:06:26Z

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← Older revision		Revision as of 00:06, 11 December 2013
Line 1:		Line 1:
	The ~~individual who wrote the article~~ is ~~called Jayson Hirano~~ and ~~he completely digs~~ that ~~title~~. ~~To perform lacross~~ is the ~~thing I adore most~~ of ~~all~~. ~~For many years she~~'s been ~~living~~ in ~~Kentucky but her husband desires them~~ to ~~transfer~~. ~~Distributing production~~ is ~~how he makes a residing~~.<br><br>~~Visit my web page~~ - ~~accurate psychic predictions~~ - [http://si.~~dgmensa~~.~~org~~/xe/~~index~~.~~php?document_srl=48014&mid=c0102 visit~~ the ~~next website page~~],		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.

			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>,

			:<math> \|y^2 - x^3\| > C\sqrt{\|x\|}.</math>

			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>.

			In 1965, Davenport proved an analogue of the above conjecture in the case of polynomials:
			if ''f''(''t'') and ''g''(''t'') are nonzero polynomials over '''C''' such that
			''g''(''t'')<sup>3</sup> ≠ ''f''(''t'')<sup>2</sup> in '''C'''[''t''], then

			:<math> \deg(g(t)^2 - f(t)^3) \geq \frac{1}{2}\deg f(t) + 1.</math>

			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>,

			:<math> \|y^2 - x^3\| > c(\varepsilon) x^{1/2-\varepsilon}.</math>

			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

			447884928428402042307918<sup>2</sup> - 5853886516781223<sup>3</sup> = -1641843,

			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.

			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]].

			==Notes==
			{{reflist}}

			==References==
			* {{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 }}
			* {{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}}

			==External links==
			* [http://www.math.harvard.edu/~elkies/hall.html], page of [[Noam Elkies]] on the problem.
			* [http://ijcalvo.galeon.com/hall.htm], table of ''good examples'' of ''Marshall Hall's conjecture'' by Ismael Jimenez Calvo.

			[[Category:Number theory]]
			[[Category:Conjectures]]

68.74.74.198 at 14:23, 27 August 2012

2012-08-27T14:23:27Z

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