ビール予想のソースを表示

'''ビール予想'''（Beal conjecture）とは、以下に示す[[数論]]の[[予想 (数学)|予想]]である。
{{unsolved|数学| ''A'', ''B'', ''C'', ''x'', ''y'', ''z'' が自然数であり、''x'', ''y'', ''z'' ≥ 3 であるとき、<math>A^x + B^y = C^z</math> ならば ''A'', ''B'', ''C'' は共通の素因数を持つか? }}
:''A'', ''B'', ''C'', ''x'', ''y'', ''z'' が[[自然数]]であり、かつ、''x'', ''y'', ''z'' ≥ 3 であるとき、
:: <math> A^x + B^y = C^z,</math>
:ならば、''A'', ''B'', ''C'' は共通の[[素因数]]を持つ。

言い換えると、次のようになる。

:''x'', ''y'', ''z'' を3以上の自然数とするとき、方程式 <math>A^x + B^y = C^z</math> は[[互いに素 (整数論)|互いに素]]となる自然数の解 ''A'', ''B'', ''C'' を持たない。

この予想は、1993年にアメリカ合衆国の銀行家でアマチュア数学者の[[アンドリュー・ビール]]が、[[フェルマーの最終定理]]の[[一般化]]の研究の過程で立てたものである<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>。1997年以降、ビールはこの予想の証明または[[反例]]を査読付きで発表した者に対する懸賞金を提供している<ref name="Mauldin"/>。懸賞金の額は何度か増額され、現在は100万[[アメリカ合衆国ドル|ドル]]となっている<ref name="BealPrize"/>。

この予想は、「一般化フェルマー方程式」(generalized Fermat equation)<ref name=":0">{{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}}</ref>、「モールディン予想」(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>、「タイデマン＝ザギエ予想」(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}}</ref>と呼ばれることもある<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>。

==例==
例を挙げると、式 <math>3^3 + 6^3 = 3^5</math> の各項の底は公約数が 3 、<math>7^3 + 7^4 = 14^3</math> は 7 、<math>2^n + 2^n = 2^{n+1}</math> は 2 である。実際、この方程式は底が共通因数を持つ解を無限に多く持ち、上の3つの例の一般化を含めて、それぞれ
:<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>

:<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>
と表せる。

さらに、各解（互いに素となる底の有無にかかわらず）に対して、指数の組が同じで、互いに素でない底の組が増えていく解が無限にある。即ち、解に対して

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

であり、加えて

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

である。ここで、

:<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>

である。

ビール予想を解くには、必ず3つの項が含まれ、その全てが[[多冪数#一般化|3-多冪数]]、すなわち、全ての素因数の指数が少なくとも3である数となることが必要となる。このような互いに素となる3-多冪数を含む和は無限にあることが知られているが<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>、それは稀である。最小の例は次の2つである。
:<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>

ビール予想の特徴は、3つの項がそれぞれ1つの冪乗で表現できることを要求していることである。

==他の予想との関係==
[[フェルマーの最終定理]]は、自然数  ''A'', ''B'', ''C'' に対して、<math>A^n + B^n = C^n</math> に ''n'' > 2 の解がないことを示している。もしフェルマーの最終定理に解が存在するならば、全ての公約数を除けば、互いに素となる自然数の解 ''A'', ''B'', ''C''が存在することになる。従って、フェルマーの最終定理は、''x'' = ''y'' = ''z'' に限定されたビール予想の[[特殊な場合]]と見ることができる。

[[フェルマー＝カタラン予想]]は、''A'', ''B'', ''C'', ''x'', ''y'', ''z'' が自然数であり、''A'', ''B'', ''C'' が互いに素である場合、<math>\frac{1}{x}+\frac{1}{y}+\frac{1}{z}<1</math> を満足するとき、<math> A^x +B^y = C^z</math> は有限個の解しか持たないというものである。ビール予想は、「全てのフェルマー＝カタラン予想の解は、2を指数として使用する」と言い換えることができる。

[[ABC予想]]は、ビール予想の反例が[[高々 (数学)|高々]]有限個であることを意味する。
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==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 (and thus the case [[Greatest common divisor|gcd]](''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}}</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}}</ref>
* 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.<ref>{{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>
* 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^3 + y^9 = z^2|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|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"/>
* The case (''x'', ''y'', ''z'') = (2, 3, 10) and all its permutations were proven by David 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|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.<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}}</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 one non-Catalan solution, which doesn't contradict Beal conjecture, by Nils Bruin in 2003.<ref name=":1" />
* The case (''x'', ''y'', ''z'') = (2, 4, ''n'') and all its permutations were proven for ''n'' ≥ 6 by Michael Bennet, [[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^2 + b^6 = c^n|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) and all its permutations were 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 $s^2+y^{2p} = \alpha^3$|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" />
* 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^3 + y^4 + z^5 = 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|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^n + y^n = z^m|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^{2n} + y^{2n} = z^5|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^{2ℓ} + y^{2m} = z^p|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|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^3 + b^3 = c^p|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}}</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|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>
-->

==懸賞金==
この予想を立てた銀行家のアンドリュー・ビールは、発表された証明または反例に対して懸賞金を提供している。懸賞金の額は、1997年の創設時は5千米ドルで、10年かけて5万米ドルまで引き上げた後<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>、100万米ドルまで引き上げられた<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>。

この懸賞金は、[[アメリカ数学会]]が信託し<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>、AMS会長が任命するビール賞委員会(BPC)が監督している<ref>{{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 |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|accessdate=2021-11-25}}</ref>。
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==Variants==
The counterexamples <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>
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==関連項目==
*[[オイラー予想]]
*[[:en:Jacobi–Madden equation|Jacobi–Madden equation]]
*[[:en:Prouhet–Tarry–Escott problem|Prouhet–Tarry–Escott problem]]
*[[タクシー数]]
*[[:en:Pythagorean quadruple|Pythagorean quadruple]]
*[[分散コンピューティング]]
*[[Berkeley Open Infrastructure for Network Computing|BOINC]]

==脚注==
{{Reflist|colwidth=30em}}

==外部リンク==
* [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

{{デフォルトソート:ひいるよそう}}
[[Category:ディオファントス方程式]]
[[Category:予想]]
[[Category:数学のオープンプロブレム]]
[[Category:数学に関する記事]]