# Beal conjecture

(Redirected from Beal's conjecture)

The Beal conjecture is the following conjecture in number theory:

If
${\displaystyle A^{x}+B^{y}=C^{z},}$
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.

## References

1. ^ "Beal Conjecture". American Mathematical Society. Retrieved 21 August 2016.
2. ^ "Beal Conjecture". Bealconjecture.com. Retrieved 2014-03-06.
3. ^ a b 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.
4. ^ a b "Beal Prize". Ams.org. Retrieved 2014-03-06.
5. ^ Bennett, Michael A.; Chen, Imin; Dahmen, Sander R.; Yazdani, Soroosh (June 2014). "Generalized Fermat Equations: A Miscellany" (PDF). Simon Fraser University. Retrieved 1 October 2016.
6. ^ "Mauldin / Tijdeman-Zagier Conjecture". Prime Puzzles. Retrieved 1 October 2016.
7. ^ a b Elkies, Noam D. (2007). "The ABC's of Number Theory" (PDF). The Harvard College Mathematics Review. 1 (1).
8. ^ Michel Waldschmidt (2004). "Open Diophantine Problems". Moscow Mathematics. 4: 245–305.
9. ^ a b Crandall, Richard; Pomerance, Carl (2000). Prime Numbers: A Computational Perspective. Springer. p. 417. ISBN 978-0387-25282-7.
10. ^ 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.
11. ^ Wacław Sierpiński, Pythagorean triangles, Dover, 2003, p. 55 (orig. Graduate School of Science, Yeshiva University, 1962).
12. ^ "Billionaire Offers $1 Million to Solve Math Problem | ABC News Blogs – Yahoo". Gma.yahoo.com. 2013-06-06. Retrieved 2014-03-06. 13. ^ a b c d Frits Beukers (January 20, 2006). "The generalized Fermat equation" (PDF). Staff.science.uu.nl. Retrieved 2014-03-06. 14. ^ Poonen, Bjorn; Schaefer, Edward F.; Stoll, Michael (2005). "Twists of X(7) and primitive solutions to x2 + y3 = z7". Duke Mathematical Journal. 137: 103–158. arXiv:. doi:10.1215/S0012-7094-07-13714-1. 15. ^ Brown, David (2009). "Primitive Integral Solutions to x2 + y3 = z10". arXiv: [math.NT]. 16. ^ "The Diophantine Equation" (PDF). Math.wisc.edu. Retrieved 2014-03-06. 17. ^ Siksek, Samir; Stoll, Michael (2013). "The Generalised Fermat Equation x2 + y3 = z15". Archiv der Mathematik. 102: 411–421. arXiv: [math.NT]. doi:10.1007/s00013-014-0639-z. 18. ^ Dahmen, Sander R.; Siksek, Samir (2013). "Perfect powers expressible as sums of two fifth or seventh powers". arXiv: [math.NT]. 19. ^ 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. 20. ^ Norvig, Peter. "Beal's Conjecture: A Search for Counterexamples". Norvig.com. Retrieved 2014-03-06. 21. ^ Walter Hickey (5 June 2013). "If You Can Solve This Math Problem, Then A Texas Banker Will Give You$1 Million". Business Insider. Retrieved 8 July 2016.
22. ^ "\$1 Million Math Problem: Banker D. Andrew Beal Offers Award To Crack Conjecture Unsolved For 30 Years". International Science Times. 5 June 2013. Retrieved 8 July 2016.
23. ^ "Neglected Gaussians". Mathpuzzle.com. Retrieved 2014-03-06.