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{{DISPLAYTITLE:''abc'' conjecture}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
The '''''abc'' conjecture''' (also known as '''Oesterlé–Masser conjecture''') is a [[conjecture]] in [[number theory]], first proposed by [[Joseph Oesterlé]] and [[David Masser]] in 1985. The conjecture is stated in terms of three positive integers, ''a'', ''b'' and ''c'' (whence comes the name), which have no common factor and satisfy ''a'' + ''b'' = ''c''. If ''d'' denotes the product of the distinct [[prime factor]]s of ''abc'', the conjecture essentially states that ''d'' is rarely much smaller than ''c''.


Although there is no obvious strategy for resolving the problem, it has already become well known for the number of [[#Some consequences|interesting consequences]] it entails. Many famous conjectures and theorems in number theory would [[#Some consequences|follow immediately]] from the ''abc'' conjecture. {{harvtxt|Goldfeld|1996}} described the ''abc'' conjecture as "the most important unsolved problem in [[Diophantine analysis]]".
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{{unsolved|mathematics|
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Are there for every ''ε'' > 0, only finitely many triples of [[coprime]] [[positive integer]]s


<center>''a''&nbsp;+&nbsp;''b''&nbsp;&#61;&nbsp;''c''</center>
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


such that
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


<center>''c''&ensp;&gt;&ensp;''d''<sup>&ensp;(1+''ε'')</sup></center>
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


where ''d'' denotes the product of the distinct prime factors of ''abc''?}}
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


==Formulations==
==Demos==
For a [[positive integer]] ''n'', the [[radical of an integer|radical]] of ''n'', denoted rad(''n''), is the product of the distinct [[prime factor]]s of ''n''. For example


* rad(16) = rad(2<sup>4</sup>) = 2,
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
* rad(17) = 17,
* rad(18) = rad(2·3<sup>2</sup>) = 2·3 = 6.


If ''a'', ''b'', and ''c'' are [[coprime]]<ref name="Ref_">Note that when it is given that ''a''&nbsp;+&nbsp;''b''&nbsp;=&nbsp;''c'', coprimeness of ''a'',&nbsp;''b'',&nbsp;''c'' implies [[pairwise coprime]]ness of ''a'',&nbsp;''b'',&nbsp;''c''. So in this case, it does not matter which concept we use.</ref> positive integers such that ''a''&nbsp;+&nbsp;''b''&nbsp;=&nbsp;''c'', it turns out that
"usually" ''c''&nbsp;<&nbsp;rad(''abc''). The ''abc conjecture'' deals with the exceptions. Specifically, it states that for every ε>0 there exist only finitely many triples (''a'',''b'',''c'') of positive coprime integers with ''a''&nbsp;+&nbsp;''b''&nbsp;=&nbsp;''c'' such that
:<math>c>\operatorname{rad}(abc)^{1+\varepsilon}.</math>


An equivalent formulation states that for any ''ε'' > 0, there exists a constant ''K'' such that, for all triples of coprime positive integers (''a'', ''b'', ''c'') satisfying ''a'' + ''b'' = ''c'', the inequality
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:<math>c < K \cdot \operatorname{rad}(abc)^{1+\varepsilon}</math>
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A third formulation of the conjecture involves the ''quality''
*[[Inputtypes|Inputtypes (private Wikis only)]]
''q''(''a'', ''b'', ''c'') of the
*[[Url2Image|Url2Image (private Wikis only)]]
triple (''a'', ''b'', ''c''), defined by:
==Bug reporting==
 
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
: <math> q(a, b, c) = \frac{ \log(c) }{ \log( \operatorname{rad}( abc ) ) }. </math>
 
For example
 
* ''q''(4, 127, 131) = log(131) / log(rad(4·127·131)) = log(131) / log(2·127·131) = 0.46820...
* ''q''(3, 125, 128) = log(128) / log(rad(3·125·128)) = log(128) / log(30) = 1.426565...
 
A typical triple (''a'', ''b'', ''c'') of coprime positive integers with ''a'' + ''b'' = ''c'' will have ''c''&nbsp;<&nbsp;rad(''abc''), i.e. ''q''(''a'', ''b'', ''c'') < 1. Triples with ''q'' > 1 such as in the second example are rather special, they consist of numbers divisible by high powers of small prime numbers.
 
The ''abc conjecture'' states that, for any ''ε'' > 0, there exist only finitely many triples (''a'', ''b'', ''c'') of coprime positive integers with ''a'' + ''b'' = ''c'' such that ''q''(''a'', ''b'', ''c'') > 1 + ''ε''.
 
Whereas it is known that there are infinitely many triples (''a'', ''b'', ''c'') of coprime positive integers with ''a'' + ''b'' = ''c'' such that ''q''(''a'', ''b'', ''c'') > 1, the conjecture predicts that only finitely many of those have ''q'' > 1.01 or ''q'' > 1.001 or even ''q'' > 1.0001, etc.
 
==Examples of triples with small radical==
The condition that ε &gt; 0 is necessary for the truth of the conjecture, as there exist infinitely many triples ''a'', ''b'', ''c'' with rad(''abc'') &lt; ''c''. For instance, such a triple may be taken as
:''a'' = 1
:''b'' = 2<sup>6''n''</sup> − 1
:''c'' = 2<sup>6''n''</sup>.
As ''a'' and ''c'' together contribute only a factor of two to the radical, while ''b'' is divisible by 9, rad(''abc'') &lt; 2''c''/3 for these examples. By replacing the exponent 6''n'' by other exponents forcing ''b'' to have larger square factors, the ratio between the radical and ''c'' may be made arbitrarily large. Another triple with a particularly small radical was found by Eric Reyssat {{harv|Lando|Zvonkin|2004|p=137}}:
:''a'' = 2:
:''b'' = 3<sup>10</sup> 109 = 6436341
:''c'' = 23<sup>5</sup> = 6436343
:rad(''abc'') = 15042.
 
==Some consequences==
The conjecture has not been proven, but it has a large number of interesting consequences. These include both known results, and conjectures for which it gives a [[conditional proof]].
* [[Thue–Siegel–Roth theorem]] on diophantine approximation of algebraic numbers
* [[Fermat's Last Theorem]] for all sufficiently large exponents (proven in general by [[Andrew Wiles]])
* The [[Mordell conjecture]] {{harv|Elkies|1991}}
* The [[Erdős–Woods number|Erdős–Woods conjecture]] except for a finite number of counterexamples {{harv|Langevin|1993}}
* The existence of infinitely many [[Wieferich prime|non-Wieferich primes]] {{harv|Silverman|1988}}
* The weak form of [[Marshall Hall's conjecture]] on the separation between squares and cubes of integers {{harv|Nitaj|1996}}
* The [[Fermat–Catalan conjecture]], a generalization of Fermat's last theorem concerning powers that are sums of powers {{harv|Pomerance|2008}}
* The [[Dirichlet L-function|L function]] ''L''(''s'',(−''d''/.)) formed with the [[Legendre symbol]], has no [[Siegel zero]] (this consequence actually requires a uniform version of the abc conjecture in number fields, not only the abc conjecture as formulated above for rational integers)
* ''P''(''x'') has only finitely many perfect powers for [[integer|integral]] ''x'' for ''P'' a [[polynomial]] with at least three simple zeros.<ref name="Ref_a">http://www.math.uu.nl/people/beukers/ABCpresentation.pdf</ref>
* A generalization of [[Tijdeman's theorem]]
* It is equivalent to the Granville–Langevin conjecture
* It is equivalent to the [[modified Szpiro conjecture]].
* {{harvtxt|Dąbrowski|1996}} has shown that the abc conjecture implies  that  ''n''! + ''A''= ''k''<sup>2</sup> has only finitely many solutions for any given integer ''A''.
 
While the first group of these have now been proven, the abc conjecture itself remains of interest, because of its numerous links with deep questions in number theory.
 
== Theoretical results ==
It remains unknown whether ''c'' can be [[upper bound]]ed by a near-linear function of the radical of ''abc'', as the abc conjecture states, or even whether it can be bounded by a [[polynomial]] of rad(''abc''). However, [[exponential function|exponential]] bounds are known. Specifically, the following bounds have been proven:
 
:<math>c < \exp{(K_1  \operatorname{rad}(abc)^{15}) } </math> {{harv|Stewart|Tijdeman|1986}},
 
:<math>c < \exp{ (K_2  \operatorname{rad}(abc)^{2/3+\varepsilon}) } </math> {{harv|Stewart|Yu|1991}}, and
 
:<math>c < \exp{ (K_3  \operatorname{rad}(abc)^{1/3+\varepsilon}) } </math> {{harv|Stewart|Yu|2001}}.
 
In these bounds, ''K''<sub>1</sub> is a [[Constant (mathematics)|constant]] that does not depend on ''a'', ''b'', or ''c'', and ''K''<sub>2</sub> and ''K''<sub>3</sub> are constants that depend on ε (in an [[effectively computable]] way) but not on ''a'', ''b'', or ''c''. The bounds apply to any triple for which ''c''&nbsp;>&nbsp;2.
 
==Computational results==
 
In 2006, the Mathematics Department of [[Leiden University]] in the Netherlands, together with the Dutch [[Kennislink]] science institute, launched the [[ABC@Home]] project, a [[grid computing]] system which aims to discover additional triples ''a'', ''b'', ''c'' with rad(''abc'') &lt; ''c''. Although no finite set of examples or counterexamples can resolve the abc conjecture, it is hoped that patterns in the triples discovered by this project will lead to insights about the conjecture and about number theory more generally.
 
{| class="wikitable sortable collapsible" border="1" style="text-align:right;"
|+ Distribution of triples with ''q'' > 1<ref name="Ref_d">{{Citation |url=http://www.rekenmeemetabc.nl/?item=h_stats |title=Synthese resultaten |work=RekenMeeMetABC.nl |accessdate=January 1, 2011 }} {{nl icon}}.</ref>
|-
! scope="col" | &nbsp;
! scope="col" | ''q'' > 1
! scope="col" | ''q'' > 1.05
! scope="col" | ''q'' > 1.1
! scope="col" | ''q'' > 1.2
! scope="col" | ''q'' > 1.3
! scope="col" | ''q'' > 1.4
|-
! scope="row" | ''c'' < 10<sup>2</sup>
| 6 || 4 || 4 || 2 || 0 || 0
|-
! scope="row" | ''c'' < 10<sup>3</sup>
| 31 || 17 || 14 || 8 || 3 || 1
|-
! scope="row" | ''c'' < 10<sup>4</sup>
| 120 || 74 || 50 || 22 || 8 || 3
|-
! scope="row" | ''c'' < 10<sup>5</sup>
| 418 || 240 || 152 || 51 || 13 || 6
|-
! scope="row" | ''c'' < 10<sup>6</sup>
| 1.268 || 667 || 379 || 102 || 29 || 11
|-
! scope="row" | ''c'' < 10<sup>7</sup>
| 3.499 || 1.669 || 856 || 210 || 60 || 17
|-
! scope="row" | ''c'' < 10<sup>8</sup>
| 8.987 || 3.869 || 1.801 || 384 || 98 || 25
|-
! scope="row" | ''c'' < 10<sup>9</sup>
| 22.316 || 8.742 || 3.693 || 706 || 144 || 34
|-
! scope="row" | ''c'' < 10<sup>10</sup>
| 51.677 || 18.233 || 7.035 || 1.159 || 218 || 51
|-
! scope="row" | ''c'' < 10<sup>11</sup>
| 116.978 || 37.612 || 13.266 || 1.947 || 327 || 64
|-
! scope="row" | ''c'' < 10<sup>12</sup>
| 252.856 || 73.714 || 23.773 || 3.028 || 455 || 74
|-
! scope="row" | ''c'' < 10<sup>13</sup>
| 528.275 || 139.762 || 41.438 || 4.519 || 599 || 84
|-
! scope="row" | ''c'' < 10<sup>14</sup>
| 1.075.319 || 258.168 || 70.047 || 6.665 || 769 || 98
|-
! scope="row" | ''c'' < 10<sup>15</sup>
| 2.131.671 || 463.446 || 115.041 || 9.497 || 998 || 112
|-
! scope="row" | ''c'' < 10<sup>16</sup>
| 4.119.410 || 812.499 || 184.727 || 13.118 || 1.232 || 126
|-
! scope="row" | ''c'' < 10<sup>17</sup>
| 7.801.334 || 1.396.909 || 290.965 || 17.890 || 1.530 || 143
|-
! scope="row" | ''c'' < 10<sup>18</sup>
| 14.482.059 || 2.352.105 || 449.194 || 24.013 || 1.843 || 160
|-
|}
 
{{As of|2012|06}}, ABC@Home has found 23.1 million triples, and its present goal is to obtain a complete list of all ABC triples (a,b,c) with c no more than 10<sup>20</sup>.<ref name="Ref_c">{{Citation |url=http://abcathome.com/data/ |title=Data collected sofar |work=ABC@Home |accessdate=June 9, 2012 }}</ref>
 
{| class="wikitable sortable collapsible" border="1"
|+ Highest quality triples<ref>{{cite web |url=http://www.math.leidenuniv.nl/~desmit/abc/index.php?set=1 |title=100 unbeaten triples |work=Reken mee met ABC |date=2010-11-07 }}</ref>
|-
! scope="col" | &nbsp;
! scope="col" | ''q''
! scope="col" | ''a''
! scope="col" | ''b''
! scope="col" | ''c''
! scope="col" class="unsortable" | Discovered by
|-
! scope="row" | 1
| 1.6299 || 2 || 3<sup>10</sup>​109 || 23<sup>5</sup> || Eric Reyssat
|-
! scope="row" | 2
| 1.6260 || 11<sup>2</sup> || 3<sup>2​</sup>5<sup>6</sup>​7<sup>3</sup> || 2<sup>21</sup>​23 || Benne de Weger
|-
! scope="row" | 3
| 1.6235 || 19·1307 || 7·​29<sup>2​</sup>31<sup>8</sup> || 2<sup>8</sup>​3<sup>22</sup>​5<sup>4</sup> || Jerzy Browkin, Juliusz Brzezinski
|-
! scope="row" | 4
| 1.5444 || 7<sup>2​</sup>41<sup>2</sup>​311<sup>3</sup> || 11<sup>16</sup>​13<sup>2</sup>​79 || 2·​3<sup>3</sup>​5<sup>23</sup>​953 || Abderrahmane Nitaj
|-
! scope="row" | 5
| 1.4805 || 5<sup>22​</sup>79·​45949 || 3<sup>2</sup>​13<sup>18</sup>​61<sup>3</sup> || 2<sup>23</sup>​17<sup>4</sup>​251<sup>2</sup>​1733<sup>3</sup> || Frank Rubin
|}
 
where the ''quality'' ''q''(''a'', ''b'', ''c'') of the triple (''a'', ''b'', ''c''), defined by:
 
: <math> q(a, b, c) = \frac{ \log(c) }{ \log( \operatorname{rad}( abc ) ) }. </math>
 
==Refined forms and generalizations==
A stronger inequality proposed in 1996 by [[Alan Baker (mathematician)|Alan Baker]] states that in the [[inequality (mathematics)|inequality]], one can replace rad(''abc'') by
 
:ε<sup>−ω</sup>rad(''abc''),
 
where ω is the total number of distinct primes dividing ''a'', ''b'' and ''c'' {{Harv|Bombieri|Gubler|2006|p=404}}. A related conjecture of [[Andrew Granville]] states that on the [[Left-hand side and right-hand side of an equation|RHS]] we could also put
 
:O(rad(''abc'') Θ(rad(''abc'')))
 
where Θ(''n'') is the number of integers up to ''n'' divisible only by primes dividing ''n''.
 
{{harvtxt|Browkin|Brzeziński|1994}} formulated the ''n''-conjecture—a version of the ''abc'' conjecture involving <math>n>2</math> integers.
 
==See also==
*[[Mason–Stothers theorem]], an analogous statement for [[polynomial]]s.
 
==Notes==
{{reflist|colwidth=45em}}
 
== References ==
{{Refbegin}}
*{{cite book | first1=Enrico | last1=Bombieri | authorlink1=Enrico Bombieri | first2=Walter | last2=Gubler | title=Heights in Diophantine Geometry | series=New Mathematical Monographs | volume=4 | publisher=[[Cambridge University Press]] | year=2006 | isbn=978-0-521-71229-3 | zbl=1130.11034 | doi=10.2277/0521846153 }}
*{{Cite journal |authorlink=Jerzy Browkin |first=Jerzy |last=Browkin |first2=Juliusz |last2=Brzeziński | title=Some remarks on the ''abc''-conjecture | journal=Math. Comp. | volume=62 | pages=931–939 | year=1994 | doi=10.2307/2153551 | jstor=2153551 | issue=206 |ref=harv }}
* {{cite book |last=Browkin |first=Jerzy |chapter=The ''abc''-conjecture |editor1-last=Bambah |editor1-first=R. P. |editor2-last=Dumir |editor2-first=V. C. |editor3-last=Hans-Gill |editor3-first=R. J. |year=2000 |title=Number Theory |series=Trends in Mathematics |location=Basel |publisher=Birkhäuser |isbn=3-7643-6259-6 |pages=75–106 |ref=harv  }}
*{{Cite journal |first=Andrzej |last=Dąbrowski |title=On the diophantine equation <math>x!+A=y^2</math> | journal=Nieuw Archief voor Wiskunde, IV. |volume=14 |pages=321–324 |year=1996 |ref=harv }}
*{{Cite journal |last=Elkies |first=N. D. |authorlink=Noam Elkies |title=ABC implies Mordell |journal=Intern. Math. Research Notices |volume=7 |year=1991 |pages=99–109 |doi=10.1155/S1073792891000144 |issue=7 |ref=harv }}
*{{cite journal |last=Goldfeld |first=Dorian |authorlink=Dorian M. Goldfeld |year=1996 |title=Beyond the last theorem |journal=[[Math Horizons]] |issue=September |pages=26–34 |ref=harv }}
* {{cite book |editor1-last=Gowers |editor1-first=Timothy |editor1-link=Timothy Gowers |editor2-last=Barrow-Green |editor2-first=June |editor3-last=Leader |editor3-first=Imre |title=[[The Princeton Companion to Mathematics]] |year=2008 |publisher=Princeton University Press |location=Princeton |isbn=978-0-691-11880-2 |pages=361–362, 681 |ref=harv  }}
* {{cite book |first=Richard K. |last=Guy |authorlink=Richard K. Guy |title=Unsolved Problems in Number Theory |publisher=[[Springer-Verlag]] |location=Berlin |year=2004 |isbn=0-387-20860-7 |ref=harv }}
* {{cite book |last=Lando |first=Sergei K. |first2=Alexander K. |last2=Zvonkin |title=Graphs on Surfaces and Their Applications |publisher=Springer-Verlag |work=Encyclopaedia of Mathematical Sciences: Lower-Dimensional Topology II |volume=141 |year=2004 |isbn=3-540-00203-0 |ref=harv }}
*{{Cite journal |last=Langevin |first=M. |year=1993 |title={{lang|fr|Cas d'égalité pour le théorème de Mason et applications de la conjecture ''abc''}} |journal=Comptes rendus de l'Académie des sciences |volume=317 |issue=5 |pages=441–444 |doi= |ref=harv }} {{fr icon}}
*{{Cite journal |last=Nitaj |first=Abderrahmane |title={{lang|fr|La conjecture ''abc''}} |journal=Enseign. Math. |volume=42 |issue=1–2 |pages=3–24 |year=1996 |ref=harv }} {{fr icon}}
*{{Cite book |last=Pomerance |first=Carl |authorlink=Carl Pomerance |chapter=Computational Number Theory |title=The Princeton Companion to Mathematics |location= |publisher=Princeton University Press |year=2008 |pages=361–362 |ref=harv }}
*{{Cite journal |last=Silverman |first=Joseph H. |year=1988 |title=Wieferich's criterion and the ''abc''-conjecture |journal=Journal of Number Theory |volume=30 |issue=2 |pages=226–237 |doi=10.1016/0022-314X(88)90019-4 | zbl=0654.10019  |ref=harv }}
*{{Cite journal |last=Stewart |first=C. L. |authorlink=Cameron Leigh Stewart |last2=Tijdeman |first2=R. |authorlink2=Robert Tijdeman |year=1986 |title=On the Oesterlé-Masser conjecture |journal=Monatshefte für Mathematik |volume=102 |issue=3 |pages=251–257 |doi=10.1007/BF01294603 |ref=harv }}
*{{Cite journal |last=Stewart |first=C. L. |authorlink2=Kunrui Yu |first2=Kunrui |last2=Yu |year=1991 |title=On the ''abc'' conjecture |journal=[[Mathematische Annalen]] |volume=291 |issue=1 |pages=225–230 |doi=10.1007/BF01445201 |ref=harv }}
*{{Cite journal |last=Stewart |first=C. L. |first2=Kunrui |last2=Yu |year=2001 |title=On the ''abc'' conjecture, II |journal=[[Duke Mathematical Journal]] |volume=108 |issue=1 |pages=169–181 |doi=10.1215/S0012-7094-01-10815-6 |ref=harv }}
{{Refend}}
 
==External links==
* [http://abcathome.com/ ABC@home] [[Distributed Computing]] project called [[ABC@Home]].
* [http://bit-player.org/2007/easy-as-abc Easy as ABC]: Easy to follow, detailed explanation by Brian Hayes.
* {{MathWorld | urlname=abcConjecture | title=abc Conjecture}}
* Abderrahmane Nitaj's [http://www.math.unicaen.fr/~nitaj/abc.html ABC conjecture home page]
* Bart de Smit's [http://www.math.leidenuniv.nl/~desmit/abc/ ABC Triples webpage]
* http://www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf
* [http://www.maa.org/mathland/mathtrek_12_8.html The amazing ABC conjecture]
* [http://www.thehcmr.org/issue1_1/elkies.pdf The ABC's of Number Theory] by Noam D. Elkies
 
[[Category:Conjectures]]
[[Category:Number theory]]
 
[[bn:Abc অনুমান]]
[[de:Abc-Vermutung]]
[[es:Conjetura abc]]
[[fr:Conjecture abc]]
[[it:Congettura abc]]
[[ja:ABC予想]]
[[hu:Abc-sejtés]]
[[nl:ABC-vermoeden]]
[[pl:Hipoteza ABC]]
[[fi:Abc-konjektuuri]]
[[tr:Abc sanısı]]
[[vi:Giả định abc]]
[[zh:Abc猜想]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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