Beal Conjecture

Statement:

If \(A^x+B^y=C^z\) where \(A, B, C, x, y, z\) are non-zero integers with \(x, y, z \geq 3 \) then \(A, B, C \) have a common prime factor.

-Equivalently-

The equation \( A^x+B^y=C^z \) has no solutions in the non-zero integers and pairwise coprime integers \(A, B, C\) provided \(x, y, z \geq 3 \).

Author	Date Published	Prize	Paid By
Andrew Beal	1993	$1,000,000	Andrew Beal (AMS)

Partial Results:

The following cases have all been shown to not contradict the Beal Conjecture:
• \( (x,y,z) \geq 3 \) [1] Andrew Wiles in 1994
• \( (x,y,z)=(2,3,7) \) [2] Bjorn Poonen, Edward F. Schaefer, and Michael Stoll in 2005
• \( (x,y,z)=(2,3,8) \) [3] Nils Bruin in 2003
• \( (x,y,z)=(2,3,9) \) [4] Nils Bruin in 2003
• \( (x,y,z)=(2,3,10) \) [5] David Brown in 2009
• \( (x,y,z)=(2,3,11) \) [6] Freitas, Naskręcki and Stoll
• \( (x,y,z)=(2,3,15) \) [7] Samir Siksek and Michael Stoll in 2013
• \( (x,y,z)=(2,4,4) \) Fermat (1640's) and Euler (1738)
• \( (x,y,z)=(2,4,5) \) [8] Nils Bruin in 2003
• \( (x,y,z)=(2,4,n) \) for \( n \geq 6 \) [9] Michael Bennet, Jordan Ellenberg, and Nathan Ng in 2009
• \( (x,y,z)=(2,6,n) \) [10] Michael Bennett and Imin Chen in 2011 and by Bennett, Chen, Dahmen and Yazdani in 2014
• \( (x,y,z)=(2,2n,3) \) for \( 3 \leq n \leq 10^7 \) except for \(7\) and various modulo congruences. [11] Bennett, Chen, Dahmen and Yazdani
• \( (x,y,z)=(2,2n,9),(2,2n,10),(2,2n,15) \) for \(n \geq 2\) \) [12] Bennett, Chen, Dahmen and Yazdani in 2014
• \( (x,y,z)=(3,3,n) \) for \( 3 \leq n \leq 10^9 \) and various modulo congruences when \(n\) is prime. [13] Frits Beukers in 2006
• \( (x,y,z)=(3,4,5) \) [14] Siksek and Stoll in 2011
• \( (x,y,z)=(3,5,5) \) [15] Bjorn Poonen in 1998
• \( (x,y,z)=(3, 6, n) \) for \( n \geq 3 \) [16] Bennett, Chen, Dahmen and Yazdani in 2014
• \( (x,y,z)=(4, 2n, 3) \) for \( n \geq 2 \) [16] Bennett, Chen, Dahmen and Yazdani in 2014
• \( (x,y,z)=(5, 5, 7), (5, 5, 19), (7, 7, 5) \) [17] Sander R. Dahmen and Samir Siksek in 2013
• \( (x,y,z)=(n, n, 2) \) for \( n \geq 4 \)[18] Darmon and Merel in 1995
• \( (x,y,z)=(n, n, 3) \) for \( n \geq 3 \)[18] Édouard Lucas, Bjorn Poonen, and Darmon and Merel
• \( (x,y,z)=(2n, 2n, 5) \) for \( n \geq 2 \)[18] Bennett in 2006
• \( (x,y,z)=(2l, 2m, n) \) for \( l, m \geq 5 \) primes and \(n=3, 5, 7, 11\) [19] Anni and Siksek
• \( (x,y,z)=(3l, 3m, n) \) for \( l,m \geq 2 \) and \(n \geq 3 \) [20] Kraus
• The Darmon–Granville theorem uses Faltings's theorem to show that for every specific choice of exponents \( (x, y, z)\), there are at most finitely many coprime solutions for \( (A, B, C)\) [21] Darmon and Granville
• The impossibility of the case A = 1 or B = 1 is implied by Catalan's conjecture [22] Preda Mihăilescu in 2002
• There are an infinite number of primitive Pythagorean triples that cannot satisfy the Beal equation [23] L. Jesmanowicz, J. Jozefiak, Chao Ko
• A series of numerical searches for counterexamples to Beal's conjecture excluded all possible solutions having each of \(x, y, z ≤ 7\) and each of \(A, B, C ≤ 250,000\), as well as possible solutions having each of \(x, y, z ≤ 100\) and each of \(A, B, C ≤ 10,000\). [24] Peter Norvig

Related Problems:

ABC Conjecture

Fermat-Catalan Conjecture

Lander, Parkin, and Selfridge conjecture

References:

[1] Wiles, Andrew. “Modular Elliptic Curves and Fermat's Last Theorem.” Annals of Mathematics, vol. 141, no. 3, 1995, pp. 443–551. JSTOR, www.jstor.org/stable/2118559. Accessed 24 May 2021.

[2] Poonen, Bjorn (1998). "Some diophantine equations of the form \(x^n + y^n = z^m\)". Acta Arithmetica (in Polish). 86 (3): 193–205. doi:10.4064/aa-86-3-193-205. ISSN 0065-1036.

[3] Bruin, Nils (2003-01-09). "Chabauty methods using elliptic curves". Journal für die reine und angewandte Mathematik (Crelles Journal). 2003 (562). doi:10.1515/crll.2003.076. ISSN 0075-4102.

[4] Bruin, Nils (2005-03-01). "The primitive solutions to \(x^3 + y^9 = z^2\)". Journal of Number Theory. 111 (1): 179–189. arXiv:math/0311002. doi:10.1016/j.jnt.2004.11.008. ISSN 0022-314X.

[5] Brown, David (2009). "Primitive Integral Solutions to \(x2 + y3 = z10\)". arXiv:0911.2932

[6] Freitas, Nuno; Naskręcki, Bartosz; Stoll, Michael (January 2020). "The generalized Fermat equation with exponents 2, 3, n". Compositio Mathematica. 156 (1): 77–113. doi:10.1112/S0010437X19007693. ISSN 0010-437X.

[7] Siksek, Samir; Stoll, Michael (2013). "The Generalised Fermat Equation \(x2 + y3 = z15\)". Archiv der Mathematik. 102 (5): 411–421. arXiv:1309.4421. doi:10.1007/s00013-014-0639-z.

[8] Bruin, Nils (2005-03-01). "The primitive solutions to \(x^3 + y^9 = z^2\)". Journal of Number Theory. 111 (1): 179–189. arXiv:math/0311002. doi:10.1016/j.jnt.2004.11.008. ISSN 0022-314X.

[9] "The Diophantine Equation" (PDF). Math.wisc.edu. Retrieved 2014-03-06.

[10] Bennett, Michael A.; Chen, Imin (2012-07-25). "Multi-Frey ℚ-curves and the Diophantine equation \(a^2 + b^6 = c^n\)". Algebra & Number Theory. 6 (4): 707–730. doi:10.2140/ant.2012.6.707. ISSN 1944-7833.

[11] Chen, Imin (2007-10-23). "On the equation \(s^2+y^{2p} = \alpha^3\)". Mathematics of Computation. 77 (262): 1223–1228. doi:10.1090/S0025-5718-07-02083-2. ISSN 0025-5718.

[12] 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.

[13] Frits Beukers (January 20, 2006). "The generalized Fermat equation" (PDF). Staff.science.uu.nl. Retrieved 2014-03-06.

[14] Siksek, Samir; Stoll, Michael (2012). "Partial descent on hyperelliptic curves and the generalized Fermat equation x^3 + y^4 + z^5 = 0". Bulletin of the London Mathematical Society. 44 (1): 151–166. arXiv:1103.1979. doi:10.1112/blms/bdr086. ISSN 1469-2120.

[15] Poonen, Bjorn (1998). "Some diophantine equations of the form x^n + y^n = z^m". Acta Arithmetica (in Polish). 86 (3): 193–205. doi:10.4064/aa-86-3-193-205. ISSN 0065-1036.

[16] 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.

[17] Dahmen, Sander R.; Siksek, Samir (2013). "Perfect powers expressible as sums of two fifth or seventh powers". arXiv:1309.4030

[18] H. Darmon and L. Merel. Winding quotients and some variants of Fermat’s Last Theorem, J. Reine Angew. Math. 490 (1997), 81–100.

[19] Anni, Samuele; Siksek, Samir (2016-08-30). "Modular elliptic curves over real abelian fields and the generalized Fermat equation \(x^{2ℓ} + y^{2m} = z^p\)". Algebra & Number Theory. 10 (6): 1147–1172. arXiv:1506.02860. doi:10.2140/ant.2016.10.1147. ISSN 1944-7833.

[20] Kraus, Alain (1998-01-01). "Sur l'équation \(a^3 + b^3 = c^p\)". Experimental Mathematics. 7 (1): 1–13. doi:10.1080/10586458.1998.10504355. ISSN 1058-6458.

[21] Darmon, H.; Granville, A. (1995). "On the equations \(zm = F(x, y) and Axp + Byq = Czr\)". Bulletin of the London Mathematical Society. 27 (6): 513–43. doi:10.1112/blms/27.6.513.

[22] Mihăilescu, Preda (2005). Reflection, Bernoulli Numbers and the Proof of Catalan's Conjecture. European Congress of Mathematics. Stockholm, Sweden: European Mathematical Society. pp. 325–340. doi:10.4171/009-1/21. MR 2185753

[23] Wacław Sierpiński, Pythagorean Triangles, Dover, 2003, p. 55 (orig. Graduate School of Science, Yeshiva University, 1962).

[24] Norvig, Peter. "Beal's Conjecture: A Search for Counterexamples". Norvig.com. Retrieved 2014-03-06.

Keywords:

Algebraic Number Theory; Beal Conjecture; Diophantine Equations