Beal's Conjecture

A generalization of Fermat's last theorem which states that if , where , , , , , and  are any positive integers with , then , , and  have a common factor. The conjecture was announced in Mauldin (1997), and a cash prize of  has been offered for its proof or a counterexample (Castelvecchi 2013).

This conjecture is more properly known as the Tijdeman-Zagier conjecture (Elkies 2007).

REFERENCES:

Brun, V. "Über hypothesesenbildungen." Arc. Math. Naturvidenskab 34, 1-14, 1914.

Castelvecchi, D. "Mathematics Prize Ups the Ante to $1 Million." June 4, 2013. https://blogs.nature.com/news/2013/06/mathematics-prize-ups-the-ante-to-1-million.html.

Darmon, H. and Granville, A. "On the Equations  and ." Bull. London Math. Soc. 27, 513-543, 1995.

Elkies, N. "The ABCs of Number Theory." Harvard Math. Rev. 1, 64-76, 2007.

Mauldin, R. D. "A Generalization of Fermat's Last Theorem: The Beal Conjecture and Prize Problem." Not. Amer. Math. Soc. 44, 1436-1437, 1997.

Mauldin, R. D. "The Beal Conjecture and Prize." https://www.math.unt.edu/~mauldin/beal.html.