Beal conjecture

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Short description: Mathematical conjecture

The Beal conjecture is the following conjecture in number theory:

Question, Web Fundamentals.svg Unsolved problem in mathematics:
If [math]\displaystyle{ A^x + B^y = C^z }[/math] where A, B, C, x, y, z are positive integers and x, y, z are ≥ 3, do A, B, and C have a common prime factor?
(more unsolved problems in mathematics)
If
[math]\displaystyle{ A^x +B^y = C^z, }[/math]
where A, B, C, x, y, and z are positive integers with x, y, z ≥ 3, then A, B, and C have a common prime factor.

Equivalently,

The equation [math]\displaystyle{ A^x + B^y = C^z }[/math] has no solutions in positive integers and pairwise coprime integers A, B, C if x, y, z ≥ 3.

The conjecture was formulated in 1993 by Andrew Beal, a banker and amateur mathematician, while investigating generalizations of Fermat's Last Theorem.[1][2] Since 1997, Beal has offered a monetary prize for a peer-reviewed proof of this conjecture or a counterexample.[3] The value of the prize has increased several times and is currently $1 million.[4]

In some publications, this conjecture has occasionally been referred to as a generalized Fermat equation,[5] the Mauldin conjecture,[6] and the Tijdeman-Zagier conjecture.[7][8][9]

Related examples

To illustrate, the solution [math]\displaystyle{ 3^3 + 6^3 = 3^5 }[/math] has bases with a common factor of 3, the solution [math]\displaystyle{ 7^3 + 7^4 = 14^3 }[/math] has bases with a common factor of 7, and [math]\displaystyle{ 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]\displaystyle{ 3^{3n}+[2(3^{n})]^{3}=3^{3n+2}, \quad\quad n \ge 1; }[/math]
[math]\displaystyle{ [b(a^{n}-b^{n})^{k}]^{n}+(a^{n}-b^{n})^{kn+1}=[a(a^{n}-b^{n})^{k}]^{n}, \quad\quad a \gt b, \quad b \ge 1, \quad k \ge 1, \quad n \ge 3; }[/math]

and

[math]\displaystyle{ [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]\displaystyle{ A_1^{x} + B_1^{y} = C_1^{z} }[/math]

we additionally have

[math]\displaystyle{ A_n^{x}+B_n^{y} = C_n^{z}; }[/math] [math]\displaystyle{ n \ge 2 }[/math]

where

[math]\displaystyle{ A_{n} = (A_{n-1}^{yz+1}) (B_{n-1}^{yz }) (C_{n-1}^{yz }) }[/math]
[math]\displaystyle{ B_{n} = (A_{n-1}^{xz }) (B_{n-1}^{xz+1}) (C_{n-1}^{xz }) }[/math]
[math]\displaystyle{ 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 3-powerful numbers, 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;[10] however, such sums are rare. The smallest two examples are:

[math]\displaystyle{ \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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}\lt 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. All cases of the form (2, 3, n) or (2, n, 3) have the solution 23 + 1n = 32 which is referred below as the Catalan solution.

  • The case x = y = z ≥ 3 (and thus the case gcd(x, y, z) ≥ 3) is Fermat's Last Theorem, proven to have no solutions by Andrew Wiles in 1994.[11]
  • 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.[12]
  • The case (x, y, z) = (2, 3, 8) and all its permutations were proven to have only one non-Catalan solution, which doesn't contradict Beal conjecture, by Nils Bruin in 2003.[13]
  • 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.[14][15][9]
  • 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.[16]
  • 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.[17]
  • The case (x, y, z) = (2, 3, 15) and all its permutations were proven by Samir Siksek and Michael Stoll in 2013.[18]
  • 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 here and another here)
  • The case (x, y, z) = (2, 4, 5) and all its permutations are known to have only one non-Catalan solution, which doesn't contradict Beal conjecture, by Nils Bruin in 2003.[14]
  • 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.[19]
  • 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.[20][5]
  • The case (x, y, z) = (2, 2n, 3) and all its permutations were proven for 3 ≤ n ≤ 107 except n = 7 and various modulo congruences when n is prime to have no non-Catalan solution by Bennett, Chen, Dahmen and Yazdani.[21][5]
  • The cases (x, y, z) = (2, 2n, 9), (2, 2n, 10), (2, 2n, 15) and all their permutations were proven for n ≥ 2 by Bennett, Chen, Dahmen and Yazdani in 2014.[5]
  • The case (x, y, z) = (3, 3, n) and all its permutations have been proven for 3 ≤ n ≤ 109 and various modulo congruences when n is prime.[15]
  • The case (x, y, z) = (3, 4, 5) and all its permutations were proven by Siksek and Stoll in 2011.[22]
  • The case (x, y, z) = (3, 5, 5) and all its permutations were proven by Bjorn Poonen in 1998.[23]
  • 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.[5]
  • The case (x, y, z) = (2n, 3, 4) and all its permutations were proven for n ≥ 2 by Bennett, Chen, Dahmen and Yazdani in 2014.[5]
  • 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.[24]
  • 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.[25][23]
  • The cases (x, y, z) = (n, n, 3) and all its permutations were proven for n ≥ 3 by Édouard Lucas, Bjorn Poonen, and Darmon and Merel.[25]
  • The case (x, y, z) = (2n, 2n, 5) and all its permutations were proven for n ≥ 2 by Bennett in 2006.[26]
  • The case (x, y, z) = (2l, 2m, n) and all its permutations were proven for l, m ≥ 5 primes and n = 3, 5, 7, 11 by Anni and Siksek.[27]
  • The case (x, y, z) = (2l, 2m, 13) and all its permutations were proven for l, m ≥ 5 primes by Billerey, Chen, Dembélé, Dieulefait, Freitas.[28]
  • The case (x, y, z) = (3l, 3m, n) is direct for l, m ≥ 2 and n ≥ 3 from work by Kraus.[29]
  • 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).[30][7]: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.[31]
  • 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.[32]
  • If A, B are odd and x, y are even, Beal's conjecture has no counterexample.[33]
  • 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]\displaystyle{ ax^m+by^n = z^r }[/math].[34]

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,[3] but has since raised it to US $1,000,000.[4]

The American Mathematical Society (AMS) holds the $1 million prize in a trust until the Beal conjecture is solved.[35] It is supervised by the Beal Prize Committee (BPC), which is appointed by the AMS president.[36]

Variants

The counterexamples [math]\displaystyle{ 7^3 + 13^2 = 2^9 }[/math] and [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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 integers. After a prize of $50 was offered for a counterexample, Fred W. Helenius provided [math]\displaystyle{ (-2+i)^3 + (-2-i)^3 = (1+i)^4 }[/math].[37]

See also

References

  1. ↑ "Beal Conjecture". American Mathematical Society. https://www.ams.org/profession/prizes-awards/ams-supported/beal-conjecture. 
  2. ↑ "Beal Conjecture". Bealconjecture.com. http://www.bealconjecture.com/. Retrieved 2014-03-06. 
  3. ↑ 3.0 3.1 R. Daniel Mauldin (1997). "A Generalization of Fermat's Last Theorem: The Beal Conjecture and Prize Problem". Notices of the AMS 44 (11): 1436–1439. https://www.ams.org/notices/199711/beal.pdf. 
  4. ↑ 4.0 4.1 "Beal Prize". Ams.org. https://www.ams.org/profession/prizes-awards/ams-supported/beal-prize. Retrieved 2014-03-06. 
  5. ↑ 5.0 5.1 5.2 5.3 5.4 5.5 Bennett, Michael A.; Chen, Imin; Dahmen, Sander R.; Yazdani, Soroosh (June 2014). "Generalized Fermat Equations: A Miscellany". Simon Fraser University. http://people.math.sfu.ca/~ichen/pub/BeChDaYa.pdf. 
  6. ↑ "Mauldin / Tijdeman-Zagier Conjecture". Prime Puzzles. http://www.primepuzzles.net/puzzles/puzz_559.htm. 
  7. ↑ 7.0 7.1 Elkies, Noam D. (2007). "The ABC's of Number Theory". The Harvard College Mathematics Review 1 (1). http://dash.harvard.edu/bitstream/handle/1/2793857/Elkies%20-%20ABCs%20of%20Number%20Theory.pdf?sequence=2. 
  8. ↑ Michel Waldschmidt (2004). "Open Diophantine Problems". Moscow Mathematical Journal 4: 245–305. doi:10.17323/1609-4514-2004-4-1-245-305. 
  9. ↑ 9.0 9.1 Crandall, Richard; Pomerance, Carl (2000). Prime Numbers: A Computational Perspective. Springer. p. 417. ISBN 978-0387-25282-7. https://archive.org/details/primenumberscomp00cran. 
  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. ↑ "Billionaire Offers $1 Million to Solve Math Problem | ABC News Blogs – Yahoo". Gma.yahoo.com. 2013-06-06. http://gma.yahoo.com/blogs/abc-blogs/billionaire-offers-1-million-solve-math-problem-153508422.html. Retrieved 2014-03-06. 
  12. ↑ 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. doi:10.1215/S0012-7094-07-13714-1. Bibcode: 2005math......8174P. 
  13. ↑ Bruin, Nils (2003-01-09). "Chabauty methods using elliptic curves". Journal für die reine und angewandte Mathematik 2003 (562). doi:10.1515/crll.2003.076. ISSN 0075-4102. 
  14. ↑ 14.0 14.1 Bruin, Nils (2005-03-01). "The primitive solutions to x^3 + y^9 = z^2" (in en). Journal of Number Theory 111 (1): 179–189. doi:10.1016/j.jnt.2004.11.008. ISSN 0022-314X. http://www.sciencedirect.com/science/article/pii/S0022314X04002471. 
  15. ↑ 15.0 15.1 Frits Beukers (January 20, 2006). "The generalized Fermat equation". Staff.science.uu.nl. http://www.staff.science.uu.nl/~beuke106/Fermatlectures.pdf. Retrieved 2014-03-06. 
  16. ↑ Brown, David (2009). "Primitive Integral Solutions to x2 + y3 = z10". arXiv:0911.2932 [math.NT].
  17. ↑ Freitas, Nuno; Naskręcki, Bartosz; Stoll, Michael (January 2020). "The generalized Fermat equation with exponents 2, 3, n" (in en). Compositio Mathematica 156 (1): 77–113. doi:10.1112/S0010437X19007693. ISSN 0010-437X. https://www.cambridge.org/core/journals/compositio-mathematica/article/generalized-fermat-equation-with-exponents-2-3-n/DAB951488A355980D5144CB78D6678AF. 
  18. ↑ Siksek, Samir; Stoll, Michael (2013). "The Generalised Fermat Equation x2 + y3 = z15". Archiv der Mathematik 102 (5): 411–421. doi:10.1007/s00013-014-0639-z. 
  19. ↑ "The Diophantine Equation". Math.wisc.edu. https://www.math.wisc.edu/~ellenber/BeElNgdraftFINAL.pdf. Retrieved 2014-03-06. 
  20. ↑ Bennett, Michael A.; Chen, Imin (2012-07-25). "Multi-Frey [math]\displaystyle{ \mathbb{Q} }[/math]-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. https://msp.org/ant/2012/6-4/p04.xhtml. 
  21. ↑ Chen, Imin (2007-10-23). "On the equation $s^2+y^{2p} = \alpha^3$" (in en). Mathematics of Computation 77 (262): 1223–1228. doi:10.1090/S0025-5718-07-02083-2. ISSN 0025-5718. https://www.ams.org/journal-getitem?pii=S0025-5718-07-02083-2. 
  22. ↑ Siksek, Samir; Stoll, Michael (2012). "Partial descent on hyperelliptic curves and the generalized Fermat equation x^3 + y^4 + z^5 = 0" (in en). Bulletin of the London Mathematical Society 44 (1): 151–166. doi:10.1112/blms/bdr086. ISSN 1469-2120. 
  23. ↑ 23.0 23.1 Poonen, Bjorn (1998). "Some diophantine equations of the form x^n + y^n = z^m" (in pl). Acta Arithmetica 86 (3): 193–205. doi:10.4064/aa-86-3-193-205. ISSN 0065-1036. 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. 
  24. ↑ Dahmen, Sander R.; Siksek, Samir (2013). "Perfect powers expressible as sums of two fifth or seventh powers". arXiv:1309.4030 [math.NT].
  25. ↑ 25.0 25.1 H. Darmon and L. Merel. Winding quotients and some variants of Fermat's Last Theorem, J. Reine Angew. Math. 490 (1997), 81–100.
  26. ↑ Bennett, Michael A. (2006). "The equation x^{2n} + y^{2n} = z^5". Journal de Théorie des Nombres de Bordeaux 18 (2): 315–321. doi:10.5802/jtnb.546. ISSN 1246-7405. https://jtnb.centre-mersenne.org/article/JTNB_2006__18_2_315_0.pdf. 
  27. ↑ 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" (in en). Algebra & Number Theory 10 (6): 1147–1172. doi:10.2140/ant.2016.10.1147. ISSN 1944-7833. http://msp.org/ant/2016/10-6/p01.xhtml. 
  28. ↑ Billerey, Nicolas; Chen, Imin; Dembélé, Lassina; Dieulefait, Luis; Freitas, Nuno (2019-03-05). "Some extensions of the modular method and Fermat equations of signature (13, 13, n)". arXiv:1802.04330 [math.NT].
  29. ↑ 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. 
  30. ↑ 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. 
  31. ↑ Wacław Sierpiński, Pythagorean Triangles, Dover, 2003, p. 55 (orig. Graduate School of Science, Yeshiva University, 1962).
  32. ↑ Norvig, Peter. "Beal's Conjecture: A Search for Counterexamples". Norvig.com. http://norvig.com/beal.html. Retrieved 2014-03-06. 
  33. ↑ "Sloane's A261782 (see the Theorem and its proof in the comment from May 08 2021)". OEIS Foundation. https://oeis.org/A261782. 
  34. ↑ Rahimi, Amir M. (2017). "An Elementary Approach to the Diophantine Equation [math]\displaystyle{ ax^m+by^n = z^r }[/math] Using Center of Mass". Missouri J. Math. Sci. 29 (2): 115–124. doi:10.35834/mjms/1513306825. 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. 
  35. ↑ Walter Hickey (5 June 2013). "If You Can Solve This Math Problem, Then A Texas Banker Will Give You $1 Million". Business Insider. http://www.businessinsider.com/beale-conjecture-1-million-dollar-prize-2013-6. 
  36. ↑ "$1 Million Math Problem: Banker D. Andrew Beal Offers Award To Crack Conjecture Unsolved For 30 Years". International Science Times. 5 June 2013. http://www.isciencetimes.com/articles/5338/20130605/1-million-math-problem-banker-d-andrew.htm. 
  37. ↑ "Neglected Gaussians". Mathpuzzle.com. http://www.mathpuzzle.com/Gaussians.html. Retrieved 2014-03-06. 

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