Beal's conjecture

Beal's conjecture is a conjecture in number theory:

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.

Equivalently,

There are no solutions to the above equation in positive integers A, B, C, x, y, z with A, B, and C being pairwise coprime and all of x, y, z being greater than 2.

Banker Andrew Beal formulated this conjecture in 1993 while investigating generalizations of Fermat's last theorem.[1] It has been claimed that the same conjecture was formulated independently by Robert Tijdeman and Don Zagier.[2] While more commonly known as the "Beal conjecture", it has also been referred to as the Tijdeman–Zagier conjecture.[3][4][5]

In the 1950s, L. Jesmanowicz and Chao Ko considered a potential class of solutions to the equation, namely those with A, B, C also forming a Pythagorean triple.[6]

References

1. ^ "Beal Conjecture". Bealconjecture.com. Retrieved 2014-03-06.
2. Frits Beukers (January 20, 2006). "The generalized Fermat equation" (PDF). Staff.science.uu.nl. Retrieved 2014-03-06.
3. ^ Elkies, Noam D. (2007). "The ABC's of Number Theory" (PDF). The Harvard College Mathematics Review 1 (1).
4. ^ Michel Waldschmidt (2004). "Open Diophantine Problems". Moscow Mathematics 4: 245–305.
5. ^ a b Crandall, Richard; Pomerance, Carl (2000). Prime Numbers: A Computational Perspective. Springer. p. 417. ISBN 978-0387-25282-7.
6. ^ Wacław Sierpiński, Pythagorean triangles, Dover, 2003, p. 55 (orig. Graduate School of Science, Yeshiva University, 1962).
7. ^ 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.
8. ^ "Beal Prize". Ams.org. Retrieved 2014-03-06.
9. ^ Nitaj, Abderrahmane (1995). "On A Conjecture of Erdos on 3-Powerful Numbers". Bulletin of the London Mathematical Society 27 (4): 317–318. doi:10.1112/blms/27.4.317.
10. ^ "Billionaire Offers \$1 Million to Solve Math Problem | ABC News Blogs – Yahoo". Gma.yahoo.com. 2013-06-06. Retrieved 2014-03-06.
11. ^ Poonen, Bjorn; Schaefer, Edward F.; Stoll, Michael (2005). "Twists of X(7) and primitive solutions to x2 + y3 = z7". arXiv:math/0508174v1.
12. ^ Brown, David (2009). "Primitive Integral Solutions to x2 + y3 = z10". arXiv:0911.2932 [math.NT].
13. ^ Siksek, Samir; Stoll, Michael (2013). "The Generalised Fermat Equation x2 + y3 = z15". arXiv:1309.4421 [math.NT].
14. ^ "The Diophantine Equation" (PDF). Math.wisc.edu. Retrieved 2014-03-06.
15. ^ Dahmen, Sander R.; Siksek, Samir (2013). "Perfect powers expressible as sums of two fifth or seventh powers". arXiv:1309.4030 [math.NT].
16. ^ Darmon, H.; Granville, A. (1995). "On the equations zm = F(x, y) and Axp + Byq = Czr". Bulletin of the London Mathematical Society 27: 513–43.
17. ^ Norvig, Peter. "Beal's Conjecture: A Search for Counterexamples". Norvig.com. Retrieved 2014-03-06.
18. ^ "Neglected Gaussians". Mathpuzzle.com. Retrieved 2014-03-06.