The Beal conjecture is the following
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
in
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
:
:If
::
,
:where ''A'', ''B'', ''C'', ''x'', ''y'', and ''z'' are positive
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s with ''x'', ''y'', ''z'' > 2, then ''A'', ''B'', and ''C'' have a common
prime factor
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
.
Equivalently,
:The equation
has no solutions in positive integers and pairwise
coprime
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...
integers ''A, B, C'' if ''x, y, z'' > 2.
The conjecture was formulated in 1993 by
Andrew Beal, a banker and
amateur mathematician, while investigating
generalization
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteri ...
s of
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than . The cases ...
.
Since 1997, Beal has offered a monetary prize for a peer-reviewed proof of this conjecture or a
counterexample
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "student John Smith is not lazy" is a c ...
.
The value of the prize has increased several times and is currently $1 million.
In some publications, this conjecture has occasionally been referred to as a generalized Fermat equation,
the Mauldin conjecture, and the Tijdeman-Zagier conjecture.
[
]
Related examples
To illustrate, the solution has bases with a common factor of 3, the solution has bases with a common factor of 7, and 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
:
:
and
:
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
:
we additionally have
:
where
:
:
:
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; however, such sums are rare. The smallest two examples are:
:
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
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than . The cases ...
established that 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 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
. Beal's conjecture can be restated as "All Fermat–Catalan conjecture solutions will use 2 as an exponent".
The abc conjecture
ABC are the first three letters of the Latin script.
ABC or abc may also refer to:
Arts, entertainment and media Broadcasting
* Aliw Broadcasting Corporation, Philippine broadcast company
* American Broadcasting Company, a commercial American ...
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 23 + 1''n'' = 32 which is referred below as the Catalan solution.
* The case ''x'' = ''y'' = ''z'' ≥ 3 is Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than . The cases ...
, proven to have no solutions by Andrew Wiles
Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory. He is best known for Wiles's proof of Fermat's Last Theorem, proving Ferma ...
in 1994.
* 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.
* 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.
* 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.
* 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.
* 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.
* 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.
* 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
Pierre de Fermat (; ; 17 August 1601 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his d ...
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 two non-Catalan solutions, which doesn't contradict Beal conjecture, by Nils Bruin in 2003.
* 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.
* 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.
* The case (''x'', ''y'', ''z'') = (2, 2''n'', 3) was 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.
* 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.
* 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.
* The case (''x'', ''y'', ''z'') = (3, 4, 5) and all its permutations were proven by Siksek and Stoll in 2011.
* The case (''x'', ''y'', ''z'') = (3, 5, 5) and all its permutations were proven by Bjorn Poonen in 1998.
* 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.
*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.
* 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.
* 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.[H. Darmon and L. Merel. Winding quotients and some variants of Fermat's Last Theorem, J. Reine Angew. Math. 490 (1997), 81–100.]
* The cases (''x'', ''y'', ''z'') = (''n'', ''n'', 3) and all its permutations were proven for ''n'' ≥ 3 by Édouard Lucas
__NOTOC__
François Édouard Anatole Lucas (; 4 April 1842 – 3 October 1891) was a French mathematician. Lucas is known for his study of the Fibonacci sequence. The related Lucas sequences and Lucas numbers are named after him.
Biography
Luc ...
, Bjorn Poonen, and Darmon and Merel.
* The case (''x'', ''y'', ''z'') = (2''n'', 2''n'', 5) and all its permutations were proven for ''n'' ≥ 2 by Bennett in 2006.
*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.
*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.
*The case (''x'', ''y'', ''z'') = (3''l'', 3''m'', ''n'') is direct for ''l'', ''m'' ≥ 2 and ''n'' ≥ 3 from work by 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'').
* 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
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A triangle whose side lengths are a Py ...
, 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.
*Peter Norvig
Peter Norvig (born 14 December 1956) is an American computer scientist and Distinguished Education Fellow at the Stanford Institute for Human-Centered AI. He previously served as a director of research and search quality at Google. Norvig is th ...
, Director of Research at Google
Google LLC (, ) is an American multinational corporation and technology company focusing on online advertising, search engine technology, cloud computing, computer software, quantum computing, e-commerce, consumer electronics, and artificial ...
, 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.
* If ''A'', ''B'' are odd and ''x'', ''y'' are even, Beal's conjecture has no counterexample.
* 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 .
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, but has since raised it to US $1,000,000.
The American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
(AMS) holds the $1 million prize in a trust until the Beal conjecture is solved. It is supervised by the Beal Prize Committee (BPC), which is appointed by the AMS president.
Variants
The counterexamples , , and 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 ).
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 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
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf ...
s. After a prize of $50 was offered for a counterexample, Fred W. Helenius provided .
See also
*ABC conjecture
ABC are the first three letters of the Latin script.
ABC or abc may also refer to:
Arts, entertainment and media Broadcasting
* Aliw Broadcasting Corporation, Philippine broadcast company
* American Broadcasting Company, a commercial American ...
*Euler's sum of powers conjecture
In number theory, Euler's conjecture is a disproved conjecture related to Fermat's Last Theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers and greater than 1, if the sum of many th powers of positive integers ...
* Jacobi–Madden equation
* Prouhet–Tarry–Escott problem
* Taxicab number
* Pythagorean quadruple
* Sums of powers, a list of related conjectures and theorems
*Distributed computing
Distributed computing is a field of computer science that studies distributed systems, defined as computer systems whose inter-communicating components are located on different networked computers.
The components of a distributed system commu ...
* BOINC
References
External links
The Beal Prize office page
Bealconjecture.com
* {{PlanetMath , title=Beal Conjecture , urlname=bealconjecture
Mathoverflow.net discussion about the name and date of origin
of the theorem
Diophantine equations
Conjectures
Unsolved problems in number theory
Abc conjecture