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.

*The above conjecture and prize was announced in the December 1997 issue of the Notices of the American Mathematical Society. Since that time Andy Beal has increased the amount of the prize for his conjecture. The prize is now this: $1,000,000 for either a proof or a counterexample of his conjecture. The prize money is being held by the American Mathematical Society until it is awarded. In the meantime the interest is being used to fund some AMS activities and the annual Erdos Memorial Lecture.*

shinichiPost authorThe Beal Conjecture

http://www.bealconjecture.com/

shinichiPost authorTHE BEAL CONJECTURE AND PRIZE

http://www.math.unt.edu/~mauldin/beal.html

shinichiPost authorBillionaire Offers $1 Million to Solve Math Problem

by Alana Abramson

ABC News

http://abcnews.go.com/blogs/headlines/2013/06/billionaire-offers-1-million-to-solve-math-problem/

He dropped out of Baylor University, but Texas billionaire D. Andrew Beal has always been fascinated by numbers and the theories behind them.

Now Beal, who, with a net worth of $8 billion ranks 43rd on the Forbes list of U.S. billionaires, is offering a $1 million reward to anyone who can solve a math problem — now dubbed the Beal Conjecture — that Beal has been trying to solve since 1993. The long-running reward started at $5,000 in 1997, and bumped up to $100,000 in 2000, where it remained for 13 years. Now the money has hit seven figures.

It all started with Beal’s determination to solve the 350-year-old mystery of Fermat’s Last Theorem — the idea that Ax + By = Cz. Beal realized that there could only be solutions to the equation when A, B, and C have a common numerical factor.

“Others have looked at other closely related problems, but I believe Beal was the first to express it in that way,” Don McClure, the executive director of the American Mathematical Society, which announced the $1 million prize, told ABC News

Beal took his Fermat’s Last Theorem findings to R. Daniel Mauldin, then a mathematics professor at the University of North Texas, who came up with the idea to offer a prize to anyone who could prove the theory.

Jordan Ellenberg, a mathematician at the University of Wisconsin, said confirming the theory could revolutionize the field.

“Any solution to this problem would signal a real new idea and not minor progress,” he said.

Mauldin, who was part of a committee that has reviewed hundreds of proposed solutions over the years, said none ever worked.

“It’s impossible to keep up with them, and none of them fit.”

With the increase in prize money, there is also a new stipulation that the solution needs to be published in a mathematics journal.

Beal declined to speak to ABCNews.com but said in a statement that he was inspired by the reward given to Andrew Wiley in 1994 when he proved Fermat’s Last Theorem.

“I’d like to inspire young people to pursue math and science. Increasing the prize is a good way to draw attention to mathematics generally. … I hope many more young people will find themselves drawn into the wonderful world of mathematics.”

But the likelihood of solving the Beal Conjecture anytime soon seems slim.

“I’m not holding my breath,” said the American Mathematical Society’s McClure.