Beal's conjecture is a conjecture in number theory proposed by the Texas billionaire and mathematical amateur Andrew Beal. In mathematics, a conjecture is a mathematical statement which appears likely to be true, but has not been formally proven to be true under the rules of mathematical logic. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Official language(s) See: Languages of Texas Capital Austin Largest city Houston Area  Ranked 2nd  - Total 268,581 sq mi (695,622 km²)  - Width 773 miles (1,244 km)  - Length 790 miles (1,270 km)  - % water 2. ... For other meanings of mathematics or math, see mathematics (disambiguation). ... Andrew Andy Beal (born 1952) is a billionaire businessman living in Dallas, Texas. ...

While investigating generalizations of Fermat's last theorem in 1993, Andrew Beal formulated the following conjecture: Pierre de Fermat Problem II.8 in the Arithmetica of Diophantus, annotated with Fermats comment which became Fermats Last Theorem (edition of 1670). ... 1993 (MCMXCIII) was a common year starting on Friday of the Gregorian calendar and marked the Beginning of the International Decade to Combat Racism and Racial Discrimination (1993-2003). ... In mathematics, a conjecture is a mathematical statement which appears likely to be true, but has not been formally proven to be true under the rules of mathematical logic. ...

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 must have a common prime factor.

For example, 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. In fact, there are infinitely many solutions whose bases have a common factor. For example, the equation

yields a solution for all a, b, m > 3. This is not a counterexample, however, since the bases all have the factor a^m + b^m in common.

As of 2006, there are no known counterexamples. Searches have been performed out to at least 1,000 in all variables.^{[1] [2]} In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule, i. ...

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 , 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. In either case, a counterexample to Fermat's Last Theorem yields a counterexample to Beal's conjecture.

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.^[3] A Gaussian integer is a complex number whose real and imaginary part are both integers. ...

Beal has offered reward of $100,000 USD for a proof or disproof of the conjecture.^[4] The United States dollar is the official currency of the United States. ...

References

1. ^ http://www.norvig.com/beal.html
2. ^ http://www.owlnet.rice.edu/~danvk/beal.html
3. ^ http://www.mathpuzzle.com/Gaussians.html
4. ^ http://www.math.unt.edu/~mauldin/beal.html

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