Beal conjecture

Beal's conjecture is a conjecture in number theory proposed by Andrew Beal in 1993. While investigating generalizations of Fermat's last theorem in 1993, Beal formulated the following conjecture:

If

$A^{x}+B^{y}=C^{z},$

where A, B, C, x, y, and z are positive integers with x, y, z > 2, then A, B, and C have a common prime factor.

For a proof or counterexample published in a refereed journal, Beal initially offered a prize of US $5,000 in 1997, raising it to $50,000 over ten years, but has since raised it to US $1,000,000.

Examples

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

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

for any a, b > 0, m > 2. 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.

Relation to other conjectures

Fermat's Last Theorem states that the equation has no solutions, or equivalently has no solutions with A, B, and C coprime, when x = y = z > 2. This is a special case of the Beal conjecture that there are no solutions with A, B, and C coprime and x, y, and z > 2 but not necessarily equal.

Beal's conjecture can be restated as "All Fermat–Catalan conjecture solutions will use 2 as an exponent."

Partial results

In the cases below where 2 is an exponent, multiples of 2 are also proven, since a power can be squared.

• The case x = 2, y = 3, and z = 7 was proved by Bjorn Poonen, Edward F. Schaefer, and Michael Stoll in 2005.

• The case x = 2, y = 4, and z a prime number was proved by Michael Bennet, Jordan Ellenberg, and Nathan Ng in 2009.

• The case x = 2, y = 3, and z = 10 was proved by David Brown in 2009.

• The case x = y = z is of course Fermat's Last Theorem, proved by Andrew Wiles in 1994.

• The case A = 1 is implied by Catalan's conjecture, proven in 2002 by Preda Mihăilescu.

• If we accept the abc conjecture, it implies that there are at most finitely many counterexamples to Beal's conjecture.

• By computerized searching, greatly accelerated by the use of hash tables, the Beal conjecture has been verified for all values of all six variables up to 1000. So in any counterexample, at least one of the variables must be greater than 1000.

Invalid variants

The counterexamples $7^{3}+13^{2}=2^{9}$ and $1^{m}+2^{3}=3^{2}$ show that the conjecture would be false if one of the exponents were allowed to be 2. The Fermat–Catalan conjecture covers cases of this sort.

A variation of the conjecture where x, y, z (instead of A, B, C) must have a common prime factor is not true. A counterexample is $27^{4}+162^{3}=9^{7}.$

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 $(-2+i)^{3}+(-2-i)^{3}=(1+i)^{4}.$

External links

• http://www.ams.org/profession/prizes-awards/ams-supported/beal-prize The Beal Prize office page
• http://www.bealconjecture.com/
• http://www.math.unt.edu/~mauldin/beal.html
• R. Daniel Mauldin (1997). "A Generalization of Fermat's Last Theorem: The Beal Conjecture and Prize Problem" (PDF). Notices of the AMS. 44 (11): 1436–1439.
• Beal Conjecture at PlanetMath.
• http://mathoverflow.net/questions/28764/status-of-beal-tijdeman-zagier-conjecture