View source for Beal conjecture

{{Short description|Mathematical conjecture}}
The '''Beal conjecture''' is the following [[conjecture]] in [[number theory]]:
{{unsolved|mathematics|If <math>A^x + B^y = C^z</math> where ''A'', ''B'', ''C'', ''x'', ''y'', ''z'' are positive integers and ''x'', ''y'', ''z'' are > ''2'', do ''A'', ''B'', and ''C'' have a common prime factor?}}
:If

:: <math> A^x +B^y = C^z</math>,

:where ''A'', ''B'', ''C'', ''x'', ''y'', and ''z'' are positive [[integer]]s with ''x'', ''y'', ''z'' > 2, then ''A'', ''B'', and ''C'' have a common [[prime factor]].<ref>{{cite web |title=Beal Prize |url=https://www.ams.org/prizes-awards/paview.cgi?parent_id=41 |publisher=American Mathematical Society |access-date=9 February 2025}}</ref>

Equivalently,

:The equation <math>A^x + B^y = C^z</math> has no solutions in positive integers and pairwise [[coprime]] integers ''A, B, C'' if ''x, y, z'' > 2.

The conjecture was formulated in 1993 by [[Andrew Beal]], a banker and [[List of amateur mathematicians|amateur mathematician]], while investigating [[generalization]]s of [[Fermat's Last Theorem]].<ref>{{cite web| url=https://www.ams.org/profession/prizes-awards/ams-supported/beal-conjecture | title=Beal Conjecture | publisher=American Mathematical Society | access-date=21 August 2016}}</ref><ref name=BealWebsite>{{cite web|title=Beal Conjecture|url=http://www.bealconjecture.com/|publisher=Bealconjecture.com|accessdate=2014-03-06}}</ref> Since 1997, Beal has offered a monetary prize for a peer-reviewed proof of this conjecture or a [[counterexample]].<ref name="Mauldin"/> The value of the prize has increased several times and is currently $1 million.<ref name="BealPrize"/>

In some publications, this conjecture has occasionally been referred to as a generalized Fermat equation,<ref name=":0">{{cite web |last1=Bennett |first1=Michael A. |last2=Chen |first2=Imin |last3=Dahmen |first3=Sander R. |last4=Yazdani |first4=Soroosh |date=June 2014 |title=Generalized Fermat Equations: A Miscellany |url=https://personal.math.ubc.ca/~bennett/BeChDaYa-IJNT-2015.pdf |access-date=1 October 2016 |publisher=Simon Fraser University}}</ref> the Mauldin conjecture,<ref>{{cite web| url = http://www.primepuzzles.net/puzzles/puzz_559.htm | title = Mauldin / Tijdeman-Zagier Conjecture | publisher = Prime Puzzles | access-date = 1 October 2016}}</ref> and the Tijdeman-Zagier conjecture.<ref name=Elkies>{{cite journal|last=Elkies| first = Noam D. | title=The ABC's of Number Theory | journal = The Harvard College Mathematics Review | year=2007 | volume=1 | issue = 1 | url=http://dash.harvard.edu/bitstream/handle/1/2793857/Elkies%20-%20ABCs%20of%20Number%20Theory.pdf?sequence=2}}</ref><ref>{{cite journal|journal= Moscow Mathematical Journal|volume=4|year=2004|title=Open Diophantine Problems|pages=245–305|author=Michel Waldschmidt|doi=10.17323/1609-4514-2004-4-1-245-305|arxiv=math/0312440|s2cid=11845578}}</ref><ref name=PrimeNumbers/>

==Related examples==
To illustrate, the solution <math>3^3 + 6^3 = 3^5</math> has bases with a common factor of 3, the solution <math>7^3 + 7^4 = 14^3</math> has bases with a common factor of 7, and <math>2^n + 2^n = 2^{n+1}</math> has bases with a common factor of 2. Indeed the equation has infinitely many solutions where the bases share a common factor, including generalizations of the above three examples, respectively

:<math>3^{3n}+[2(3^{n})]^{3}=3^{3n+2}, \quad\quad n \ge 1;</math>

:<math>[b(a^{n}-b^{n})^{k}]^{n}+(a^{n}-b^{n})^{kn+1}=[a(a^{n}-b^{n})^{k}]^{n}, \quad\quad a > b, \quad b \ge 1, \quad k \ge 1, \quad n \ge 3;</math>

and

:<math>[a(a^{n}+b^{n})^{k}]^{n}+[b(a^{n}+b^{n})^{k}]^{n}=(a^{n}+b^{n})^{kn+1}, \quad \quad a \ge 1, \quad b \ge 1, \quad k \ge 1, \quad n \ge 3.</math>

Furthermore, for each solution (with or without coprime bases), there are infinitely many solutions with the same set of exponents and an increasing set of non-coprime bases. That is, for solution

:<math>A_1^{x} + B_1^{y} = C_1^{z}</math>

we additionally have

:<math>A_n^{x}+B_n^{y} = C_n^{z};</math> <math>n \ge 2</math>

where

:<math>A_{n} = (A_{n-1}^{yz+1}) (B_{n-1}^{yz  }) (C_{n-1}^{yz  })</math>
:<math>B_{n} = (A_{n-1}^{xz  }) (B_{n-1}^{xz+1}) (C_{n-1}^{xz  })</math>
:<math>C_{n} = (A_{n-1}^{xy  }) (B_{n-1}^{xy  }) (C_{n-1}^{xy+1})</math>

Any solutions to the Beal conjecture will necessarily involve three terms all of which are [[Powerful number#Generalization|3-powerful number]]s, i.e. numbers where the exponent of every prime factor is at least three. It is known that there are an infinite number of such sums involving coprime 3-powerful numbers;<ref>{{cite journal|title=On A Conjecture of Erdos on 3-Powerful Numbers|last = Nitaj|first=Abderrahmane|year=1995|journal=Bulletin of the London Mathematical Society|volume=27|issue=4|pages=317–318|doi=10.1112/blms/27.4.317|citeseerx = 10.1.1.24.563}}</ref> however, such sums are rare. The smallest two examples are:

:<math>\begin{align}
271^3 + 2^3\ 3^5\ 73^3 = 919^3 &= 776{,}151{,}559 \\
3^4\ 29^3\ 89^3 + 7^3\ 11^3\ 167^3 = 2^7\ 5^4\ 353^3 &= 3{,}518{,}958{,}160{,}000 \\
\end{align}</math>

What distinguishes Beal's conjecture is that it requires each of the three terms to be expressible as a single power.

==Relation to other conjectures==
[[Fermat's Last Theorem]] established that <math>A^n + B^n = C^n</math> has no solutions for ''n'' > 2 for positive integers ''A'', ''B'', and ''C''. If any solutions had existed to Fermat's Last Theorem, then by dividing out every common factor, there would also exist solutions with ''A'', ''B'', and ''C'' coprime. Hence, Fermat's Last Theorem can be seen as a [[special case]] of the Beal conjecture restricted to ''x'' = ''y'' = ''z''.

The [[Fermat–Catalan conjecture]] is that <math> A^x +B^y = C^z</math> has only finitely many solutions with ''A'', ''B'', and ''C'' being positive integers with no common prime factor and ''x'', ''y'', and ''z'' being positive integers satisfying
<math display="inline"> \frac{1}{x}+\frac{1}{y}+\frac{1}{z} < 1 </math>. Beal's conjecture can be restated as "All Fermat–Catalan conjecture solutions will use 2 as an exponent".

The [[abc conjecture]] would imply that there are at most finitely many counterexamples to Beal's conjecture.

==Partial results==
In the cases below where ''n'' is an exponent, multiples of ''n'' are also proven, since a ''kn''-th power is also an ''n''-th power. Where solutions involving a second power are alluded to below, they can be found specifically at [[Fermat–Catalan conjecture#Known solutions]]. All cases of the form (2, 3, ''n'') or (2, ''n'', 3) have the solution 2<sup>3</sup> + 1<sup>''n''</sup> = 3<sup>2</sup> which is referred below as the '''Catalan solution'''.

* The case ''x'' = ''y'' = ''z'' ≥ 3 is [[Fermat's Last Theorem]], proven to have no solutions by [[Andrew Wiles]] in 1994.<ref>{{cite web|url=http://gma.yahoo.com/blogs/abc-blogs/billionaire-offers-1-million-solve-math-problem-153508422.html|title=Billionaire Offers $1 Million to Solve Math Problem {{pipe}} ABC News Blogs – Yahoo|date=2013-06-06|publisher=Gma.yahoo.com|accessdate=2014-03-06|archive-date=2013-06-13|archive-url=https://web.archive.org/web/20130613181605/http://gma.yahoo.com/blogs/abc-blogs/billionaire-offers-1-million-solve-math-problem-153508422.html|url-status=dead}}</ref>
* The case (''x'', ''y'', ''z'') = (2, 3, 7) and all its permutations were proven to have only four non-Catalan solutions, none of them contradicting Beal conjecture, by [[Bjorn Poonen]], Edward F. Schaefer, and Michael Stoll in 2005.<ref>{{cite journal |arxiv=math/0508174|last1=Poonen|first1=Bjorn|title=Twists of ''X''(7) and primitive solutions to ''x''<sup>2</sup>&nbsp;+&nbsp;''y''<sup>3</sup>&nbsp;=&nbsp;''z''<sup>7</sup>|last2= Schaefer|first2=Edward F.|last3=Stoll|first3=Michael|year=2005|doi=10.1215/S0012-7094-07-13714-1|volume=137|journal=Duke Mathematical Journal|pages=103–158|bibcode=2005math......8174P|s2cid=2326034}}</ref>
* The case (''x'', ''y'', ''z'') = (2, 3, 8) and all its permutations were proven to have only two non-Catalan solutions, which doesn't contradict Beal conjecture, by Nils Bruin in 2003.<ref name="NB">{{Cite journal|last=Bruin|first=Nils|date=2003-01-09|title=Chabauty methods using elliptic curves|journal=Journal für die reine und angewandte Mathematik|volume=2003|issue=562|doi=10.1515/crll.2003.076|issn=0075-4102}}</ref><ref>{{Cite journal|last=Bruin|first=Nils|date=1999-09-01|title=The diophantine equations x<sup>2</sup> ± y<sup>4</sup> = ±z<sup>6</sup> and x<sup>2</sup> + y<sup>8</sup> = z<sup>3</sup>|journal=Compositio Mathematica |volume=118|issue=3|pages=305–321 |doi=10.1023/A:1001529706709|doi-broken-date=1 November 2024 |issn=0010-437X}}</ref>
* The case (''x'', ''y'', ''z'') = (2, 3, 9) and all its permutations are known to have only one non-Catalan solution, which doesn't contradict Beal conjecture, by Nils Bruin in 2003.<ref name=":1">{{Cite journal|last=Bruin|first=Nils|date=2005-03-01|title=The primitive solutions to x<sup>3</sup> + y<sup>9</sup> = z<sup>2</sup>|url=http://www.sciencedirect.com/science/article/pii/S0022314X04002471|journal=Journal of Number Theory|language=en|volume=111|issue=1|pages=179–189|doi=10.1016/j.jnt.2004.11.008|arxiv=math/0311002|s2cid=9704470|issn=0022-314X}}</ref><ref name="FB">{{cite web|url=http://www.staff.science.uu.nl/~beuke106/Fermatlectures.pdf|title=The generalized Fermat equation|author=Frits Beukers|author-link=Frits Beukers|date=January 20, 2006|publisher=Staff.science.uu.nl|accessdate=2014-03-06}}</ref><ref name="PrimeNumbers">{{cite book |title=Prime Numbers: A Computational Perspective |url=https://archive.org/details/primenumberscomp00cran |url-access=limited |last1=Crandall |first1=Richard |last2=Pomerance |first2=Carl |year=2000 |isbn=978-0387-25282-7 |publisher=Springer |page=[https://archive.org/details/primenumberscomp00cran/page/n425 417]}}</ref>
* The case (''x'', ''y'', ''z'') = (2, 3, 10) and all its permutations were proven by David Zureick-Brown in 2009 to have only the Catalan solution.<ref>{{cite arXiv |eprint=0911.2932|last1=Brown|first1=David|title=Primitive Integral Solutions to ''x''<sup>2</sup> + ''y''<sup>3</sup> = ''z''<sup>10</sup>|class=math.NT|year=2009}}</ref>
* The case (''x'', ''y'', ''z'') = (2, 3, 11) and all its permutations were proven by Freitas, Naskręcki and Stoll to have only the Catalan solution.<ref>{{Cite journal|last1=Freitas|first1=Nuno|last2=Naskręcki|first2=Bartosz|last3=Stoll|first3=Michael|date=January 2020|title=The generalized Fermat equation with exponents 2, 3, n|url=https://www.cambridge.org/core/journals/compositio-mathematica/article/generalized-fermat-equation-with-exponents-2-3-n/DAB951488A355980D5144CB78D6678AF|journal=Compositio Mathematica|language=en|volume=156|issue=1|pages=77–113|doi=10.1112/S0010437X19007693|s2cid=15030869|issn=0010-437X}}</ref>
* The case (''x'', ''y'', ''z'') = (2, 3, 15) and all its permutations were proven by Samir Siksek and Michael Stoll in 2013 to have only the Catalan solution.<ref>{{cite journal|last1=Siksek|first1=Samir|last2=Stoll|first2=Michael|year=2013|title=The Generalised Fermat Equation ''x''<sup>2</sup> + ''y''<sup>3</sup> = ''z''<sup>15</sup>|journal=Archiv der Mathematik|volume=102|issue=5|pages=411–421|arxiv=1309.4421|doi=10.1007/s00013-014-0639-z|s2cid=14582110}}</ref>
* The case (''x'', ''y'', ''z'') = (2, 4, 4) and all its permutations were proven to have no solutions by combined work of [[Pierre de Fermat]] in the 1640s and Euler in 1738. (See one proof [[Proof by infinite descent#Non-solvability of r2 + s4 = t4|here]] and another [[Fermat's right triangle theorem#Fermat's proof|here]])
* The case (''x'', ''y'', ''z'') = (2, 4, 5) and all its permutations are known to have only two non-Catalan solutions, which doesn't contradict Beal conjecture, by Nils Bruin in 2003.<ref name="NB" />
* The case (''x'', ''y'', ''z'') = (2, 4, ''n'') and all its permutations were proven for ''n'' ≥ 6 by Michael Bennett, [[Jordan Ellenberg]], and Nathan Ng in 2009.<ref>{{cite web|url=https://www.math.wisc.edu/~ellenber/BeElNgdraftFINAL.pdf|title=The Diophantine Equation|publisher=Math.wisc.edu|accessdate=2014-03-06}}</ref>
* The case (''x'', ''y'', ''z'') = (2, 6, ''n'') and all its permutations were proven for ''n'' ≥ 3 by Michael Bennett and Imin Chen in 2011 and by Bennett, Chen, Dahmen and Yazdani in 2014.<ref>{{Cite journal|last1=Bennett|first1=Michael A.|last2=Chen|first2=Imin|date=2012-07-25|title=Multi-Frey <math>\mathbb{Q}</math>-curves and the Diophantine equation a<sup>2</sup> + b<sup>6</sup> = c<sup>n</sup>|url=https://msp.org/ant/2012/6-4/p04.xhtml|journal=Algebra & Number Theory|volume=6|issue=4|pages=707–730|doi=10.2140/ant.2012.6.707|issn=1944-7833|doi-access=free}}</ref><ref name=":0" />
* The case (''x'', ''y'', ''z'') = (2, 2''n'', 3) was proven for 3 ≤ ''n'' ≤ 10<sup>7</sup> except ''n'' = 7 and various modulo congruences when ''n'' is prime to have no non-Catalan solution by Bennett, Chen, Dahmen and Yazdani.<ref>{{Cite journal|last=Chen|first=Imin|date=2007-10-23|title=On the equation s<sub>2</sub>+y<sup>2p</sup> = &alpha;<sup>3</sup> |url=https://www.ams.org/journal-getitem?pii=S0025-5718-07-02083-2|journal=Mathematics of Computation|language=en|volume=77|issue=262|pages=1223–1228|doi=10.1090/S0025-5718-07-02083-2|issn=0025-5718|doi-access=free}}</ref><ref name=":0" /><ref>{{Cite journal|last=Dahmen|first=Sander|date=2011|title=A Refined Modular Approach to the Diophantie Equation x<sup>2</sup> + y<sup>2n</sup> = z<sup>3</sup>|journal=International Journal of Number Theory|volume=7|issue=5|pages=1303–1316|doi=10.1142/S1793042111004472|arxiv=1002.0020 |issn=1793-7310}}</ref>
* The cases (''x'', ''y'', ''z'') = (2, 2''n'', 9), (2, 2''n'', 10), (2, 2''n'', 15) and all their permutations were proven for ''n'' ≥ 2 by Bennett, Chen, Dahmen and Yazdani in 2014.<ref name=":0" />
* The case (''x'', ''y'', ''z'') = (3, 3, ''n'') and all its permutations have been proven for 3 ≤ ''n'' ≤ 10<sup>9</sup> and various modulo congruences when ''n'' is prime.<ref name="FB" />
* The case (''x'', ''y'', ''z'') = (3, 4, 5) and all its permutations were proven by Siksek and Stoll in 2011.<ref>{{Cite journal|last1=Siksek|first1=Samir|last2=Stoll|first2=Michael|date=2012|title=Partial descent on hyperelliptic curves and the generalized Fermat equation x<sup>3</sup> + y<sup>4</sup> + z<sup>5</sup> = 0|journal=Bulletin of the London Mathematical Society|language=en|volume=44|issue=1|pages=151–166|doi=10.1112/blms/bdr086|arxiv=1103.1979|s2cid=12565749|issn=1469-2120}}</ref>
* The case (''x'', ''y'', ''z'') = (3, 5, 5) and all its permutations were proven by [[Bjorn Poonen]] in 1998.<ref name=":2">{{Cite journal|last=Poonen|first=Bjorn|date=1998|title=Some diophantine equations of the form x<sup>n</sup> + y<sup>n</sup> = z<sup>m</sup>|url=https://www.impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/acta-arithmetica/all/86/3/110170/some-diophantine-equations-of-the-form-x-n-y-n-z-m|journal=Acta Arithmetica|language=pl|volume=86|issue=3|pages=193–205|doi=10.4064/aa-86-3-193-205|issn=0065-1036|doi-access=free}}</ref>
* The case (''x'', ''y'', ''z'') = (3, 6, ''n'') and all its permutations were proven for ''n'' ≥ 3 by Bennett, Chen, Dahmen and Yazdani in 2014.<ref name=":0" />
*The case (''x'', ''y'', ''z'') = (2''n'', 3, 4) and all its permutations were proven for ''n'' ≥ 2 by Bennett, Chen, Dahmen and Yazdani in 2014.<ref name=":0" />
* The cases (5, 5, 7), (5, 5, 19), (7, 7, 5) and all their permutations were proven by Sander R. Dahmen and Samir Siksek in 2013.<ref>{{cite arXiv |eprint=1309.4030|last1= Dahmen|first1= Sander R.|title= Perfect powers expressible as sums of two fifth or seventh powers|last2= Siksek|first2= Samir|class= math.NT|year= 2013}}</ref>
* The cases (''x'', ''y'', ''z'') = (''n'', ''n'', 2) and all its permutations were proven for ''n'' ≥ 4 by Darmon and Merel in 1995 following work from Euler and Poonen.<ref name="DM">H. Darmon and L. Merel. Winding quotients and some variants of Fermat's Last Theorem, J. Reine Angew. Math. 490 (1997), 81–100.</ref><ref name=":2" />
* The cases (''x'', ''y'', ''z'') = (''n'', ''n'', 3) and all its permutations were proven for ''n'' ≥ 3 by [[Édouard Lucas]], [[Bjorn Poonen]], and [[Henri Darmon|Darmon]] and [[Loïc Merel|Merel]].<ref name="DM" />
* The case (''x'', ''y'', ''z'') = (2''n'', 2''n'', 5) and all its permutations were proven for ''n'' ≥ 2 by Bennett in 2006.<ref>{{Cite journal|last=Bennett|first=Michael A.|date=2006|title=The equation x<sup>2n</sup> + y<sup>2n</sup> = z<sup>5</sup>|url=https://jtnb.centre-mersenne.org/article/JTNB_2006__18_2_315_0.pdf|journal=Journal de Théorie des Nombres de Bordeaux|volume=18|issue=2|pages=315–321|doi=10.5802/jtnb.546|issn=1246-7405}}</ref>
*The case (''x'', ''y'', ''z'') = (2''l'', 2''m'', ''n'') and all its permutations were proven for ''l'', ''m'' ≥ 5 primes and ''n'' = 3, 5, 7, 11 by Anni and Siksek.<ref>{{Cite journal|last1=Anni|first1=Samuele|last2=Siksek|first2=Samir|date=2016-08-30|title=Modular elliptic curves over real abelian fields and the generalized Fermat equation x<sup>2ℓ</sup> + y<sup>2m</sup> = z<sup>p</sup>|url=http://msp.org/ant/2016/10-6/p01.xhtml|journal=Algebra & Number Theory|language=en|volume=10|issue=6|pages=1147–1172|doi=10.2140/ant.2016.10.1147|arxiv=1506.02860|s2cid=118935511|issn=1944-7833}}</ref>
*The case (''x'', ''y'', ''z'') = (2''l'', 2''m'', 13) and all its permutations were proven for ''l'', ''m'' ≥ 5 primes by Billerey, Chen, Dembélé, Dieulefait, Freitas.<ref>{{cite arXiv|last1=Billerey|first1=Nicolas|last2=Chen|first2=Imin|last3=Dembélé|first3=Lassina|last4=Dieulefait|first4=Luis|last5=Freitas|first5=Nuno|date=2019-03-05|title=Some extensions of the modular method and Fermat equations of signature (13, 13, n)|class=math.NT|eprint=1802.04330}}</ref>
*The case (''x'', ''y'', ''z'') = (3''l'', 3''m'', ''n'') is direct for ''l'', ''m'' ≥ 2 and ''n'' ≥ 3 from work by Kraus.<ref>{{Cite journal|last=Kraus|first=Alain|date=1998-01-01|title=Sur l'équation a<sup>3</sup> + b<sup>3</sup> = c<sup>p</sup>|journal=Experimental Mathematics|volume=7|issue=1|pages=1–13|doi=10.1080/10586458.1998.10504355|issn=1058-6458}}</ref>
*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'').<ref>{{cite journal |first1=H. |last1=Darmon |first2=A. |last2=Granville |title=On the equations ''z''<sup>''m''</sup> = ''F''(''x'', ''y'') and ''Ax''<sup>''p''</sup> + ''By''<sup>''q''</sup> = ''Cz''<sup>''r''</sup> |journal=Bulletin of the London Mathematical Society |volume=27 |issue=6 |pages=513–43 |year=1995|doi=10.1112/blms/27.6.513|doi-access=free }}</ref><ref name="Elkies" />{{rp|p. 64}}
* The impossibility of the case ''A'' = 1 or ''B'' = 1 is implied by [[Catalan's conjecture]], proven in 2002 by [[Preda Mihăilescu]]. (Notice ''C'' cannot be 1, or one of ''A'' and ''B'' must be 0, which is not permitted.)
*A potential class of solutions to the equation, namely those with ''A, B, C'' also forming a [[Pythagorean triple]], were considered by L. Jesmanowicz in the 1950s. J. Jozefiak proved that there are an infinite number of primitive Pythagorean triples that cannot satisfy the Beal equation. Further results are due to Chao Ko.<ref>[[Wacław Sierpiński]], ''[[Pythagorean Triangles]]'', Dover, 2003, p. 55 (orig. Graduate School of Science, Yeshiva University, 1962).</ref>
*[[Peter Norvig]], Director of Research at [[Google]], reported having conducted a series of numerical searches for counterexamples to Beal's conjecture. Among his results, he 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.<ref>{{cite web| last=Norvig| first=Peter| url=http://norvig.com/beal.html| title=Beal's Conjecture: A Search for Counterexamples|publisher=Norvig.com|accessdate=2014-03-06}}</ref>
* If ''A'', ''B'' are odd and ''x'', ''y'' are even, Beal's conjecture has no counterexample.<ref name="fourth_theorem">{{Cite web|url=https://oeis.org/A261782|title=Sloane's A261782 (see the Theorem and its proof in the comment from May 08 2021)|last=|first=|date=|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2021-06-19}}</ref>
* By assuming the validity of Beal's conjecture, there exists an upper bound for any common divisor of ''x'', ''y'' and ''z'' in the expression <math> ax^m+by^n = z^r </math>.<ref name=Rahimi>{{cite journal|author=Rahimi, Amir M.|year=2017|title=An Elementary Approach to the Diophantine Equation <math> ax^m+by^n = z^r </math> Using Center of Mass|journal=Missouri J. Math. Sci.|volume=29|issue=2|pages=115–124|doi=10.35834/mjms/1513306825|url=https://projecteuclid.org/journals/missouri-journal-of-mathematical-sciences/volume-29/issue-2/An-Elementary-Approach-to-the-Diophantine-Equation-axm--byn/10.35834/mjms/1513306825.short}}</ref>

==Prize==

For a published proof or counterexample, banker [[Andrew Beal]] initially offered a prize of US $5,000 in 1997, raising it to $50,000 over ten years,<ref name="Mauldin">{{cite journal |author=R. Daniel Mauldin |title=A Generalization of Fermat's Last Theorem: The Beal Conjecture and Prize Problem |journal=[[Notices of the AMS]] |volume=44 |issue=11 |pages=1436–1439 |year=1997 |url=https://www.ams.org/notices/199711/beal.pdf}}</ref> but has since raised it to US $1,000,000.<ref name="BealPrize">{{cite web|url=https://www.ams.org/profession/prizes-awards/ams-supported/beal-prize |title=Beal Prize |publisher=Ams.org |accessdate=2014-03-06}}</ref>

The [[American Mathematical Society]] (AMS) holds the $1 million prize in a trust until the Beal conjecture is solved.<ref>{{cite web| url=http://www.businessinsider.com/beale-conjecture-1-million-dollar-prize-2013-6 | title=If You Can Solve This Math Problem, Then A Texas Banker Will Give You $1 Million | author=Walter Hickey | date=5 June 2013 | publisher=Business Insider | access-date=8 July 2016}}</ref> It is supervised by the Beal Prize Committee (BPC), which is appointed by the AMS president.<ref>{{cite news| url=http://www.isciencetimes.com/articles/5338/20130605/1-million-math-problem-banker-d-andrew.htm | title=$1 Million Math Problem: Banker D. Andrew Beal Offers Award To Crack Conjecture Unsolved For 30 Years | date=5 June 2013 | newspaper=International Science Times |archive-url = https://web.archive.org/web/20170929232723/http://www.isciencetimes.com/articles/5338/20130605/1-million-math-problem-banker-d-andrew.htm|archive-date =29 September 2017 | last1=Times | first1=Science }}</ref>

==Variants==
The counterexamples <math>3^5 + 10^2 = 7^3</math>, <math> 7^3 + 13^2 = 2^9</math>, and <math>1^m + 2^3 = 3^2</math> show that the conjecture would be false if one of the exponents were allowed to be 2. The [[Fermat–Catalan conjecture]] is an open conjecture dealing with such cases (the condition of this conjecture is that the sum of the reciprocals is less than 1). If we allow at most one of the exponents to be 2, then there may be only finitely many solutions (except the case <math>1^m + 2^3 = 3^2</math>).

If ''A'', ''B'', ''C'' can have a common prime factor then the conjecture is not true; a classic counterexample is <math>2^{10} + 2^{10} = 2^{11}</math>.

A variation of the conjecture asserting that ''x'', ''y'', ''z'' (instead of ''A'', ''B'', ''C'') must have a common prime factor is not true. A counterexample is <math>27^4 +162^3 = 9^7,</math> in which 4, 3, and 7 have no common prime factor. (In fact, the maximum common prime factor of the exponents that is valid is 2; a common factor greater than 2 would be a counterexample to Fermat's Last Theorem.)

The conjecture is not valid over the larger domain of [[Gaussian integer]]s. After a prize of $50 was offered for a counterexample, Fred W. Helenius provided <math>(-2+i)^3 + (-2-i)^3 = (1+i)^4</math>.<ref>{{cite web|url=http://www.mathpuzzle.com/Gaussians.html |title=Neglected Gaussians |publisher=Mathpuzzle.com |accessdate=2014-03-06}}</ref>

==See also==
*[[ABC conjecture]]
*[[Euler's sum of powers conjecture]]
*[[Jacobi–Madden equation]]
*[[Prouhet–Tarry–Escott problem]]
*[[Taxicab number]]
*[[Pythagorean quadruple]]
*[[Sums of powers]], a list of related conjectures and theorems
*[[Distributed computing]]
*[[BOINC]]

==References==
{{Reflist|colwidth=30em}}

==External links==
* [https://www.ams.org/profession/prizes-awards/ams-supported/beal-prize The Beal Prize office page]
* [http://www.bealconjecture.com/ Bealconjecture.com]
* [http://www.math.unt.edu/~mauldin/beal.html Math.unt.edu]
* {{PlanetMath |title=Beal Conjecture |urlname=bealconjecture}}
* [https://mathoverflow.net/q/28764 Mathoverflow.net discussion about the name and date of origin] of the theorem

[[Category:Diophantine equations]]
[[Category:Conjectures]]
[[Category:Unsolved problems in number theory]]
[[Category:Abc conjecture]]