Beal conjecture - Revision history

Vipul: /* =Stronger facts and conjectures */

2010-08-13T17:54:42Z

=Stronger facts and conjectures

← Older revision		Revision as of 17:54, 13 August 2010
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	==Related facts==		==Related facts==

	~~===Stronger facts and conjectures==~~

	===Weaker facts and conjectures===		===Weaker facts and conjectures===

Vipul: /* Failure of slight modifications of the conjecture */

2010-08-13T17:54:21Z

Failure of slight modifications of the conjecture

← Older revision		Revision as of 17:54, 13 August 2010
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	* [[Beal conjecture fails over Gaussian integers]]		* [[Beal conjecture fails over Gaussian integers]]
	* [[Beal conjecture fails if condition of relative primality is dropped]]: In fact, there are easy-to-parametrize families of solutions.		* [[Beal conjecture fails if condition of relative primality is dropped]]: In fact, there are easy-to-parametrize families of solutions.
			* [[Analogue of Beal conjecture with relative primality condition on exponents instead of bases fails]]

	==External links==		==External links==

Vipul at 17:53, 13 August 2010

2010-08-13T17:53:27Z

← Older revision		Revision as of 17:53, 13 August 2010
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	Consider the equation:		Consider the equation:

	<math>x^l + y^m = z^n</math>.		<math>\! a^m + b^n = c^k</math>.

	The '''Beal conjecture''' (also called '''Beal's conjecture''') states the following equivalent things:		The '''Beal conjecture''' (also called '''Beal's conjecture''') states the following equivalent things:

	# This equation has no solutions for <math>x,y,z</math> pairwise relatively prime positive integers, and <math>l,m,n</math> all natural numbers greater than <math>2</math>.		# This equation has no solutions for <math>a,b,c</math> pairwise relatively prime positive integers, and <math>m,n,k</math> all natural numbers greater than <math>2</math>.
	# This equation has no solutions for <math>x,y,z</math> pairwise relatively prime integers (all nonzero) and <math>l,m,n</math> all natural numbers greater than <math>2</math>.		# This equation has no solutions for <math>a,b,c</math> pairwise relatively prime integers (all nonzero) and <math>m,n,k</math> all natural numbers greater than <math>2</math>.

	==Related facts==		==Related facts==

			===Stronger facts and conjectures==

	===Weaker facts and conjectures===		===Weaker facts and conjectures===

	* [[Fermat's last theorem]]: This is the special case <math>~~l =~~ m = n</math>. This was conjectured by Fermat and proved by Wiles, building on work by several mathematicians in between.		* [[Fermat's last theorem]]: This is the special case <math>m = n = k</math>. This was conjectured by Fermat and proved by Wiles, building on work by several mathematicians in between.

	===Other related facts===		===Other related facts===

	* [[Euler's false attempted generalization of Fermat's last theorem]]		* [[Euler's false attempted generalization of Fermat's last theorem]]
			* [[Fermat-Catalan conjecture]]: With somewhat weaker hypotheses on the exponents, it claims that there are only finitely many solutions.

	===Failure of slight modifications of the conjecture===		===Failure of slight modifications of the conjecture===

Vipul: Created page with '==History== This conjecture was made by Andrew Beal, a mathematics hobbyist, while investigating Fermat's last theorem. ==Statement== Consider the equation: $x^l + y...'$

2009-04-19T21:38:32Z

Created page with '==History== This conjecture was made by Andrew Beal, a mathematics hobbyist, while investigating Fermat's last theorem. ==Statement== Consider the equation: <math>x^l + y...'

New page

==History==

This conjecture was made by Andrew Beal, a mathematics hobbyist, while investigating [[Fermat's last theorem]].

==Statement==

Consider the equation:

<math>x^l + y^m = z^n</math>.

The '''Beal conjecture''' (also called '''Beal's conjecture''') states the following equivalent things:

# This equation has no solutions for <math>x,y,z</math> pairwise relatively prime positive integers, and <math>l,m,n</math> all natural numbers greater than <math>2</math>.
# This equation has no solutions for <math>x,y,z</math> pairwise relatively prime integers (all nonzero) and <math>l,m,n</math> all natural numbers greater than <math>2</math>.

==Related facts==

===Weaker facts and conjectures===

* [[Fermat's last theorem]]: This is the special case <math>l = m = n</math>. This was conjectured by Fermat and proved by Wiles, building on work by several mathematicians in between.

===Other related facts===

* [[Euler's false attempted generalization of Fermat's last theorem]]

===Failure of slight modifications of the conjecture===

* [[Beal conjecture fails over Gaussian integers]]
* [[Beal conjecture fails if condition of relative primality is dropped]]: In fact, there are easy-to-parametrize families of solutions.

==External links==

* [http://www.bealconjecture.com Beal Conjecture page]
* [http://www.math.unt.edu/~mauldin/beal.html Information about Beal Conjecture and prize]
* [http://www.norvig.com/beal.html Peter Norvig's explanation of a computer program for the Beal conjecture]

Beal conjecture - Revision history

Vipul: /* =Stronger facts and conjectures */

Vipul: /* Failure of slight modifications of the conjecture */

Vipul at 17:53, 13 August 2010

Vipul: Created page with '==History== This conjecture was made by Andrew Beal, a mathematics hobbyist, while investigating Fermat's last theorem. ==Statement== Consider the equation: x^l + y...'

Vipul: Created page with '==History== This conjecture was made by Andrew Beal, a mathematics hobbyist, while investigating Fermat's last theorem. ==Statement== Consider the equation: $x^l + y...'$