Beal's conjecture

Beal's conjecture is a conjecture in number theory proposed by the Texas billionaire and amateur mathematician Andrew Beal.

While investigating generalizations of Fermat's last theorem in 1993, Beal formulated the following conjecture:

If: $left.\; A^x\; +B^y\; =\; C^z\; ight.$, where $A$, $B$, $C$, $x$, $y$ and $z$ are positive integers with $x,\; y,\; z\; >\; 2$ then $A$, $B$, and $C$ must have a common prime factor.

By computerized searching, greatly accelerated by aid of modular arithmetic, this conjecture has been verified for all values of all six variables up to 1000. [ [http://www.norvig.com/beal.html Beal's Conjecture: A Search for Counterexamples ] ] So in any counterexample, at least one of the variables must be greater than 1000.

To illustrate, the solution 3³ + 6³ = 3⁵ has bases with a common factor of 3, and the solution 7⁶ + 7⁷ = 98³ has bases with a common factor of 7. Indeed the equation has infinitely many solutions, including for example

: $left\; [a\; left(a^m\; +\; b^m\; ight)\; ight]\; ^m\; +\; left\; [b\; left(a^m\; +\; b^m\; ight)\; ight]\; ^m\; =\; left(a^m+b^m\; ight)^\{m+1\}$

for any $a$, $b$, $m\; >\; 3$. But no such solution of the equation is a counterexample to the conjecture, since the bases all have the factor $a^m\; +\; b^m$ in common.

It can happen that the "exponents" are pairwise coprime, as for example in $27^4\; +162^3\; =\; 9^7$.

Beal's conjecture is a generalization of Fermat's last theorem, which corresponds to the case $x\; =\; y\; =\; z$. If $a^x\; +\; b^x\; =\; c^x$ with $x\; ge\; 3$, then either the bases are coprime or share a common factor. If they share a common factor, it can be divided out of each to yield an equation with smaller, coprime bases.

The conjecture is not valid over the larger domain of Gaussian integers. After a prize of $50 was offered for a counterexample, Fred W. Helenius provided (−2 + "i")³ + (−2 − "i")³ = (1 + "i")⁴. [ [http://www.mathpuzzle.com/Gaussians.html Neglected Gaussians ] ]

Beal has offered a prize of US$100,000 for a proof of his conjecture or a counterexample [ [http://www.math.unt.edu/~mauldin/beal.html The Beal Conjecture ] ] .

References

External links

* http://www.bealconjecture.com/
* http://www.math.unt.edu/~mauldin/beal.html
* http://www.ams.org/notices/199711/beal.pdf
* A search for counterexamples - http://www.norvig.com/beal.html
* http://planetmath.org/encyclopedia/BealsConjecture.html