Beal conjecture: Difference between revisions

Revision as of 17:53, 13 August 2010

History

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

Statement

Consider the equation:

$\!a^{m}+b^{n}=c^{k}$ .

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

This equation has no solutions for $a,b,c$ pairwise relatively prime positive integers, and $m,n,k$ all natural numbers greater than $2$ .
This equation has no solutions for $a,b,c$ pairwise relatively prime integers (all nonzero) and $m,n,k$ all natural numbers greater than $2$ .

Related facts

=Stronger facts and conjectures

Weaker facts and conjectures

Fermat's last theorem: This is the special case $m=n=k$ . 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
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

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.

@@ Line 7: / Line 7: @@
 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:
-# 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==
+===Stronger 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===
 * [[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===