词条 | Beal conjecture |
释义 |
The Beal conjecture is the following conjecture in number theory: {{unsolved|mathematics|Is the Beal conjecture true?}}If 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. Equivalently, There are no solutions to the above equation in positive integers A, B, C, x, y, z with A, B, and C being pairwise coprime and all of x, y, z being greater than 2. 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,[3] the Mauldin conjecture,[4] and the Tijdeman-Zagier conjecture.[5][6][9] Related examplesTo 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;[7] 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 conjecturesFermat's Last Theorem 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 would imply that there are at most finitely many counterexamples to Beal's conjecture. Partial resultsIn the cases below where 2 is an exponent, multiples of 2 are also proven, since a power can be squared. Similarly, where n is an exponent, multiples of n are also proven. Where solutions involving a second power are alluded to below, they can be found specifically at Fermat-Catalan conjecture#Known solutions.
PrizeFor 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,[20] but has since raised it to US $1,000,000.[21] The American Mathematical Society (AMS) holds the $1 million prize in a trust until the Beal conjecture is solved.[22] It is supervised by the Beal Prize Committee (BPC), which is appointed by the AMS president.[23] VariantsThe 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. If we allow that at most one of the exponents is 2, then there may be only finitely many solutions (except the case ). If A, B, C can have a common prime factor then the conjecture is not true; a classic counterexample is . 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 integers. After a prize of $50 was offered for a counterexample, Fred W. Helenius provided [24] See also
References1. ^{{cite web| url=http://www.ams.org/profession/prizes-awards/ams-supported/beal-conjecture | title=Beal Conjecture | publisher=American Mathematical Society | access-date=21 August 2016}} 2. ^{{cite web|title=Beal Conjecture|url=http://www.bealconjecture.com/|publisher=Bealconjecture.com|accessdate=2014-03-06}} 3. ^{{cite web | url = http://people.math.sfu.ca/~ichen/pub/BeChDaYa.pdf | title = Generalized Fermat Equations: A Miscellany | first1 = Michael A. | last1=Bennett | first2=Imin | last2=Chen | first3 = Sander R. | last3=Dahmen | first4= Soroosh | last4 = Yazdani | date = June 2014 | publisher = Simon Fraser University | access-date=1 October 2016}} 4. ^{{cite web| url = http://www.primepuzzles.net/puzzles/puzz_559.htm | title = Mauldin / Tijdeman-Zagier Conjecture | publisher = Prime Puzzles | access-date = 1 October 2016}} 5. ^1 {{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}} 6. ^{{cite journal|journal=Moscow Mathematics|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}} 7. ^{{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}} 8. ^Wacław Sierpiński, Pythagorean triangles, Dover, 2003, p. 55 (orig. Graduate School of Science, Yeshiva University, 1962). 9. ^{{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 | ABC News Blogs – Yahoo |publisher=Gma.yahoo.com |date=2013-06-06 |accessdate=2014-03-06}} 10. ^1 H. Darmon and L. Merel. Winding quotients and some variants of Fermat’s Last Theorem, J. Reine Angew. Math. 490 (1997), 81–100. 11. ^1 2 3 4 5 6 {{cite web|author=Frits Beukers|title=The generalized Fermat equation|date=January 20, 2006|url=http://www.staff.science.uu.nl/~beuke106/Fermatlectures.pdf|publisher=Staff.science.uu.nl|accessdate=2014-03-06}} 12. ^{{cite journal |arxiv=math/0508174v1|last1=Poonen|first1=Bjorn|title=Twists of X(7) and primitive solutions to x2 + y3 = z7|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}} 13. ^1 {{cite book |title=Prime Numbers: A Computational Perspective |last1=Crandall |first1=Richard |last2=Pomerance |first2=Carl |year=2000 |isbn=978-0387-25282-7 |publisher=Springer |page=417}} 14. ^{{cite arXiv |eprint=0911.2932|last1=Brown|first1=David|title=Primitive Integral Solutions to x2 + y3 = z10|class=math.NT|year=2009}} 15. ^{{cite web|url=https://www.math.wisc.edu/~ellenber/BeElNgdraftFINAL.pdf |title=The Diophantine Equation |publisher=Math.wisc.edu |accessdate=2014-03-06}} 16. ^{{cite journal |arxiv=1309.4421|last1=Siksek|first1=Samir|title=The Generalised Fermat Equation x2 + y3 = z15|last2=Stoll|first2=Michael|year=2013|doi=10.1007/s00013-014-0639-z|volume=102|issue=5|journal=Archiv der Mathematik|pages=411–421}} 17. ^{{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}} 18. ^{{cite journal |first1=H. |last1=Darmon |first2=A. |last2=Granville |title=On the equations zm = F(x, y) and Axp + Byq = Czr |journal=Bulletin of the London Mathematical Society |volume=27 |issue=6 |pages=513–43 |year=1995|doi=10.1112/blms/27.6.513}} 19. ^{{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}} 20. ^1 {{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=http://www.ams.org/notices/199711/beal.pdf}} 21. ^1 {{cite web|url=http://www.ams.org/profession/prizes-awards/ams-supported/beal-prize |title=Beal Prize |publisher=Ams.org |date= |accessdate=2014-03-06}} 22. ^{{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}} 23. ^{{cite web| 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 | publisher=International Science Times | access-date=|archiveurl = https://web.archive.org/web/20170929232723/http://www.isciencetimes.com/articles/5338/20130605/1-million-math-problem-banker-d-andrew.htm|archivedate =29 September 2017}} 24. ^{{cite web|url=http://www.mathpuzzle.com/Gaussians.html |title=Neglected Gaussians |publisher=Mathpuzzle.com |date= |accessdate=2014-03-06}} External links
4 : Diophantine equations|Conjectures|Unsolved problems in mathematics|Number theory |
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