Beal's conjecture

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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.

Billionaire banker Andrew Beal discovered this conjecture in 1993 while investigating generalizations of Fermat's last theorem.[1] It has been claimed that the same conjecture was later formulated independently by Robert Tijdeman and Don Zagier.[2] While "Beal conjecture" is the more commonly accepted reference, it has also been referred to as the "Tijdeman–Zagier conjecture" in one published article.[3] In the 1950s, L. Jesmanowicz and Chao Ko considered the same question with the added restriction that A^2+B^2=C^2.[4]

For a proof or counterexample published in a refereed journal, Beal initially offered a prize of US $5,000 in 1997, raising it to $50,000 over ten years,[5] but has since raised it to US $1,000,000.[6]

Related examples[edit]

To illustrate, the solution  3^3 + 6^3 = 3^5 has bases with a common factor of 3, the solution  7^3 + 7^4 = 14^3 has bases with a common factor of 7, and  2^n + 2^n = 2^{n+1} 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

3^{3n}+[2(3^{n})]^{3}=3^{3n+2}; n \ge1
(a^{n}-1)^{2n}+(a^{n}-1)^{2n+1}=[a(a^{n}-1)^{2}]^{n}; a \ge2, n \ge3

and

[a(a^n+b^n)]^n+[b(a^n+b^n)]^n=(a^n+b^n)^{n+1}; a \ge1, b \ge1, n \ge3

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 coprime bases. That is, for solution

A_1^{x}+B_1^{y}=C_1^{z}

we additionally have

A_{n}^{x}+B_{n}^{y}=C_{n}^{z}; n \ge2

where

A_{n}= (A_{n-1}^{yz+1}) (B_{n-1}^{yz  }) (C_{n-1}^{yz  })
B_{n}= (A_{n-1}^{xz  }) (B_{n-1}^{xz+1}) (C_{n-1}^{xz  })
C_{n}= (A_{n-1}^{xy  }) (B_{n-1}^{xy  }) (C_{n-1}^{xy+1})

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;[7] however, such sums are rare. The smallest two examples are:

\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}

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[edit]

Fermat's Last Theorem established that A^n + B^n = C^n 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  A^x +B^y = C^z 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 \frac{1}{x}+\frac{1}{y}+\frac{1}{z}<1. 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[edit]

In the cases below where 2 is an exponent, multiples of 2 are also proven, since a power can be squared.

  • The case gcd(x,y,z) > 2 is implied by Fermat's Last Theorem.
  • The case y = z = 4 has been proven for all x.[2]
  • The case (x, y, z) = (2, 3, 7) and all its permutations were proven to have only four solutions, none of them involving an even power greater than 2, by Bjorn Poonen, Edward F. Schaefer, and Michael Stoll in 2005.[9]
  • The case (x, y, z) = (2, 3, 8) and all its permutations are known to have only three solutions, none of them involving an even power greater than 2.[2]
  • The case (x, y, z) = (2, 3, 9) and all its permutations are known to have only two solutions, neither of them involving an even power greater than 2.[2][10]
  • The case (x, y, z) = (2, 3, 10) was proved by David Brown in 2009.[11]
  • The case (x, y, z) = (2, 3, 15) was proved by Samir Siksek and Michael Stoll in 2013.[12]
  • The case (x, y, z) = (2, 4, n) was proved for n ≥ 4 by Michael Bennet, Jordan Ellenberg, and Nathan Ng in 2009.[13]
  • The case (x, y, z) = (n, n, 2) has been proven for n any integer other than 3 or a 2 power.[2]
  • The case (x, y, z) = (n, n, 3) has been proven.[2]
  • The case (x, y, z) = (3, 3, n) has been proven for n equal to 4, 5, or 17 ≤ n ≤ 10000.[2]
  • The cases (5, 5, 7), (5, 5, 19) and (7, 7, 5) were proved by Sander R. Dahmen and Samir Siksek in 2013.[14]
  • Faltings' theorem implies that for every specific choice of exponents (x,y,z), there are at most finitely many solutions.[15]
  • 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.[16]

Invalid variants[edit]

The counterexamples  7^3 + 13^2 = 2^9 and 1^m + 2^3 = 3^2 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.

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 27^4 +162^3 = 9^7, 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 (-2+i)^3 + (-2-i)^3 = (1+i)^4.[17]

See also[edit]

References[edit]

  1. ^ "Beal Conjecture". Bealconjecture.com. Retrieved 2014-03-06. 
  2. ^ a b c d e f g Frits Beukers (January 20, 2006). "The generalized Fermat equation". Staff.science.uu.nl. Retrieved 2014-03-06. 
  3. ^ Elkies, Noam D. (2007). "The ABC's of Number Theory". The Harvard College Mathematics Review 1 (1). 
  4. ^ Wacław Sierpiński, Pythagorean triangles, Dover, 2003, p. 55 (orig. Graduate School of Science, Yeshiva University, 1962).
  5. ^ 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. 
  6. ^ "Beal Prize". Ams.org. Retrieved 2014-03-06. 
  7. ^ 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. 
  8. ^ "Billionaire Offers $1 Million to Solve Math Problem | ABC News Blogs – Yahoo". Gma.yahoo.com. 2013-06-06. Retrieved 2014-03-06. 
  9. ^ Poonen, Bjorn; Schaefer, Edward F.; Stoll, Michael (2005). "Twists of X(7) and primitive solutions to x2 + y3 = z7". arXiv:math/0508174v1.
  10. ^ Crandall, Richard; Pomerance, Carl (2000). Prime Numbers: A Computational Perspective. Springer. p. 417. ISBN 978-0387-25282-7. 
  11. ^ Brown, David (2009). "Primitive Integral Solutions to x2 + y3 = z10". arXiv:0911.2932 [math.NT].
  12. ^ Siksek, Samir; Stoll, Michael (2013). "The Generalised Fermat Equation x2 + y3 = z15". arXiv:1309.4421 [math.NT].
  13. ^ "The Diophantine Equation". Math.wisc.edu. Retrieved 2014-03-06. 
  14. ^ Dahmen, Sander R.; Siksek, Samir (2013). "Perfect powers expressible as sums of two fifth or seventh powers". arXiv:1309.4030 [math.NT].
  15. ^ 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. 
  16. ^ Norvig, Peter. "Beal's Conjecture: A Search for Counterexamples". Norvig.com. Retrieved 2014-03-06. 
  17. ^ "Neglected Gaussians". Mathpuzzle.com. Retrieved 2014-03-06. 

External links[edit]