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| {{DISPLAYTITLE:''abc'' conjecture}}
| | Hi there. Let me start by introducing the author, her name is Myrtle Cleary. Puerto Rico is where he's std home test usually been [http://Www.Anylabtestnow.com/franchises/hialeah-33012/ residing] std home test but she needs to transfer simply because of her family. My day job is a meter reader. One of the issues she loves most is to do aerobics and now she is trying to make cash with it.<br><br>Here is my weblog - [http://localilatinoamericanomilano.it/solid-advice-in-relation-to-yeast-infection/ http://localilatinoamericanomilano.it] |
| The '''''abc'' conjecture''' (also known as '''Oesterlé–Masser conjecture''') is a [[conjecture]] in [[number theory]], first proposed by {{harvs|txt|authorlink=Joseph Oesterlé|first=Joseph|last=Oesterlé|year=1988}} and {{harvs|txt|authorlink=David Masser|first=David|last=Masser|year= 1985}} as an integer analogue of the [[Mason–Stothers theorem]] for [[polynomial]]s. The conjecture is stated in terms of three positive integers, ''a'', ''b'' and ''c'' (hence 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 usually not much smaller than ''c''. In other words: if ''a'' and ''b'' are composed from large powers of primes, then ''c'' is usually not divisible by large powers of primes. The precise statement is given below.
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| The abc conjecture has already become well known for the number of [[#Some consequences|interesting consequences]] it entails. Many famous conjectures and theorems in number theory would 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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| Several solutions have been proposed to the ''abc'' conjecture, the most recent of which is still being evaluated by the mathematical community, though it still remains open as of January 2014.
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| ==Formulations==
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| The abc conjecture can be expressed as follows:
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| For every ε > 0, there are only finitely many triples of [[coprime]] [[positive integer]]s ''a'' + ''b'' {{=}} ''c'' such that ''c'' > ''d''<sup>1+ε</sup>, where ''d'' denotes the product of the distinct prime factors of ''abc''.
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| To illustrate the terms used, if
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| :''a'' = 16 = 2<sup>4</sup>,
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| :''b'' = 17, and
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| :''c'' = 16 + 17 = 33 = 3·11,
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| then ''d'' = 2·17·3·11 = 1122, which is greater than ''c''. Therefore, for all ε > 0, ''c'' is not greater than ''d''<sup>1+ε</sup>. According to the conjecture, most coprime triples where {{nowrap|1=''a'' + ''b'' = ''c''}} are like the ones used in this example, and for only a few exceptions is ''c'' > ''d''<sup>1+ε</sup>.
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| To add more terminology:
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| 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
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| * rad(16) = rad(2<sup>4</sup>) = 2,
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| * rad(17) = 17,
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| * rad(18) = rad(2·3<sup>2</sup>) = 2·3 = 6.
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| If ''a'', ''b'', and ''c'' are [[coprime]]<ref name="Ref_">Note that when it is given that ''a'' + ''b'' = ''c'', coprimeness of ''a'', ''b'', ''c'' implies [[pairwise coprime]]ness of ''a'', ''b'', ''c''. So in this case, it does not matter which concept we use.</ref> positive integers such that ''a'' + ''b'' = ''c'', it turns out that "usually" ''c'' < rad(''abc''). The ''abc conjecture'' deals with the exceptions. Specifically, it states that:
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| <blockquote> '''ABC Conjecture.''' For every ε > 0, there exist only finitely many triples (''a'', ''b'', ''c'') of positive coprime integers, with ''a'' + ''b'' = ''c'', such that
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| :<math>c>\operatorname{rad}(abc)^{1+\varepsilon}</math>
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| </blockquote>
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| An equivalent formulation states that:
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| <blockquote> '''ABC Conjecture II.''' For every ε > 0, there exists a constant ''K''<sub>ε</sub> such that for all triples (''a'', ''b'', ''c'') of coprime positive integers, with ''a'' + ''b'' = ''c'', the inequality
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| :<math>c < K_{\varepsilon} \cdot \operatorname{rad}(abc)^{1+\varepsilon}</math>
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| holds.</blockquote>
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| A third formulation of the conjecture involves the ''quality'', ''q''(''a'', ''b'', ''c''), of the triple (''a'', ''b'', ''c''), defined by:
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| : <math> q(a, b, c) = \frac{ \log(c) }{ \log( \operatorname{rad}( abc ) ) }</math>
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| For example,
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| * ''q''(4, 127, 131) = log(131) / log(rad(4·127·131)) = log(131) / log(2·127·131) = 0.46820...
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| * ''q''(3, 125, 128) = log(128) / log(rad(3·125·128)) = log(128) / log(30) = 1.426565...
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| A typical triple (''a'', ''b'', ''c'') of coprime positive integers with ''a'' + ''b'' = ''c'' will have ''c'' < 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 number]]s.
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| <blockquote> '''ABC Conjecture III.''' For every ''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''.</blockquote>
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| 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.
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| ==Examples of triples with small radical==
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| The condition that ε > 0 is necessary for the truth of the conjecture, as there exist infinitely many triples ''a'', ''b'', ''c'' with rad(''abc'') < ''c''. For instance, such a triple may be taken as
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| :''a'' = 1
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| :''b'' = 2<sup>6''n''</sup> − 1
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| :''c'' = 2<sup>6''n''</sup>
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| As ''a'' and ''c'' together contribute only a factor of two to the radical, while ''b'' is divisible by 9, rad(''abc'') < 2''c''/3 for these examples, if n > 1. 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 small. Specifically, replacing 6''n'' by ''p''(''p'' − 1)''n'' for an arbitrary prime ''p'' will make ''b'' divisible by ''p''<sup>2</sup>, because 2<sup>''p''(''p''−1)</sup> ≡ 1 (mod ''p''<sup>2</sup>) and 2<sup>''p''(''p''−1)</sup> − 1 will be a factor of ''b''.
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| A list of the [[#Highest quality triples|highest quality triples]] (triples with a particularly small radical relative to ''c'') is given below; the highest quality of these, with quality 1.6299, was found by Eric Reyssat {{harv|Lando|Zvonkin|2004|p=137}}:
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| :''a'' = 2
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| :''b'' = 3<sup>10</sup>·109 = 6,436,341
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| :''c'' = 23<sup>5</sup> = 6,436,343
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| :rad(''abc'') = 15042
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| ==Some consequences==
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| The abc conjecture has a large number of consequences. These include both known results (some of which have been proven separately since the conjecture has been stated), and conjectures for which it gives a [[conditional proof]]. While an earlier proof of the conjecture would have been more significant in terms of consequences, the abc conjecture itself remains of interest for the other conjectures it would prove, together with its numerous links with deep questions in number theory.
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| * [[Thue–Siegel–Roth theorem]] on diophantine approximation of algebraic numbers
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| * [[Fermat's Last Theorem]] for all sufficiently large exponents (already proven in general by [[Andrew Wiles]]) {{harv|Granville |2002}}
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| * The [[Mordell conjecture]] (already proven in general by [[Gerd Faltings]]) {{harv|Elkies|1991}}
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| * The [[Erdős–Woods number|Erdős–Woods conjecture]] except for a finite number of counterexamples {{harv|Langevin|1993}}
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| * The existence of infinitely many [[Wieferich prime|non-Wieferich primes]] {{harv|Silverman|1988}}
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| * The weak form of [[Marshall Hall's conjecture]] on the separation between squares and cubes of integers {{harv|Nitaj|1996}}
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| * The [[Fermat–Catalan conjecture]], a generalization of Fermat's last theorem concerning powers that are sums of powers {{harv|Pomerance|2008}}
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| * 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) {{harv|Granville|2000}}
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| * ''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>
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| * A generalization of [[Tijdeman's theorem]] concerning the number of solutions of ''y<sup>m</sup>'' = ''x<sup>n</sup>'' + ''k'' (Tijdeman's theorem answers the case ''k'' = 1), and Pillai's conjecture (1931) concerning the number of solutions of ''Ay<sup>m</sup>'' = ''Bx<sup>n</sup>'' + ''k''.
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| * It is equivalent to the Granville–Langevin conjecture, that if ''f'' is a square-free binary form of degree ''n'' > 2, then for every real β>2 there is a constant ''C''(''f'',β) such for all coprime integer ''x'',''y'', the radical of ''f''(''x'',''y'') exceeds ''C''.max{|''x''|,|''y''|}<sup>''n''-β</sup>.<ref>Mollin (2009)</ref><ref>Mollin (2010) p.297</ref>
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| * It is equivalent to the modified [[Szpiro conjecture]], which would yield a bound of rad(''abc'')<sup>1.2+ε</sup> {{harv|Oesterlé|1988}}.
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| * {{harvtxt|Dąbrowski|1996}} has shown that the abc conjecture implies that [[Brocard's problem|the Diophantine equation ''n''! + ''A''= ''k''<sup>2</sup>]] has only finitely many solutions for any given integer ''A''.
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| * There are ~''c''<sub>''f''</sub>''N'' positive integers ''n'' ≤ ''N'' for which ''f''(''n'')/B' is squarefree, with ''c''<sub>''f''</sub> > 0 a positive constant defined as <math>c_f = \prod_{p\ prime} x_i \left ( 1 - \frac{\omega\,\!_f (p)}{p^{2+q_p}} \right )</math>. {{harv|Granville|1998}}
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| ==Theoretical results==
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| The abc conjecture implies that ''c'' can be [[upper bound|bounded above]] by a near-linear function of the radical of ''abc''. However, [[exponential function|exponential]] bounds are known. Specifically, the following bounds have been proven:
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| :<math>c < \exp{ \left(K_1 \operatorname{rad}(abc)^{15}\right) } </math> {{harv|Stewart|Tijdeman|1986}},
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| :<math>c < \exp{ \left(K_2 \operatorname{rad}(abc)^{\frac{2}{3} + \varepsilon}\right) } </math> {{harv|Stewart|Yu|1991}}, and
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| :<math>c < \exp{ \left(K_3 \operatorname{rad}(abc)^{\frac{1}{3} + \varepsilon}\right) } </math> {{harv|Stewart|Yu|2001}}.
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| 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'' > 2.
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| ==Computational results==
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| 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'') < ''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.
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| {| class="wikitable sortable" style="text-align:right;"
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| |+ 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=October 3, 2012 }} {{nl icon}}.</ref>
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| |-
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| ! scope="col" |
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| ! scope="col" | ''q'' > 1
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| ! scope="col" | ''q'' > 1.05
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| ! scope="col" | ''q'' > 1.1
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| ! scope="col" | ''q'' > 1.2
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| ! scope="col" | ''q'' > 1.3
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| ! scope="col" | ''q'' > 1.4
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| |-
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| ! scope="row" | ''c'' < 10<sup>2</sup>
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| | 6 || 4 || 4 || 2 || 0 || 0
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| |-
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| ! scope="row" | ''c'' < 10<sup>3</sup>
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| | 31 || 17 || 14 || 8 || 3 || 1
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| |-
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| ! scope="row" | ''c'' < 10<sup>4</sup>
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| | 120 || 74 || 50 || 22 || 8 || 3
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| |-
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| ! scope="row" | ''c'' < 10<sup>5</sup>
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| | 418 || 240 || 152 || 51 || 13 || 6
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| |-
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| ! scope="row" | ''c'' < 10<sup>6</sup>
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| | 1,268 || 667 || 379 || 102 || 29 || 11
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| |-
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| ! scope="row" | ''c'' < 10<sup>7</sup>
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| | 3,499 || 1,669 || 856 || 210 || 60 || 17
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| |-
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| ! scope="row" | ''c'' < 10<sup>8</sup>
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| | 8,987 || 3,869 || 1,801 || 384 || 98 || 25
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| |-
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| ! scope="row" | ''c'' < 10<sup>9</sup>
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| | 22,316 || 8,742 || 3,693 || 706 || 144 || 34
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| |-
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| ! scope="row" | ''c'' < 10<sup>10</sup>
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| | 51,677 || 18,233 || 7,035 || 1,159 || 218 || 51
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| |-
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| ! scope="row" | ''c'' < 10<sup>11</sup>
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| | 116,978 || 37,612 || 13,266 || 1,947 || 327 || 64
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| |-
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| ! scope="row" | ''c'' < 10<sup>12</sup>
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| | 252,856 || 73,714 || 23,773 || 3,028 || 455 || 74
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| |-
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| ! scope="row" | ''c'' < 10<sup>13</sup>
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| | 528,275 || 139,762 || 41,438 || 4,519 || 599 || 84
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| |-
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| ! scope="row" | ''c'' < 10<sup>14</sup>
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| | 1,075,319 || 258,168 || 70,047 || 6,665 || 769 || 98
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| |-
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| ! scope="row" | ''c'' < 10<sup>15</sup>
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| | 2,131,671 || 463,446 || 115,041 || 9,497 || 998 || 112
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| |-
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| ! scope="row" | ''c'' < 10<sup>16</sup>
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| | 4,119,410 || 812,499 || 184,727 || 13,118 || 1,232 || 126
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| |-
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| ! scope="row" | ''c'' < 10<sup>17</sup>
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| | 7,801,334 || 1,396,909 || 290,965 || 17,890 || 1,530 || 143
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| |-
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| ! scope="row" | ''c'' < 10<sup>18</sup>
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| | 14,482,065 || 2,352,105 || 449,194 || 24,013 || 1,843 || 160
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| |-
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| |}
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| {{As of|2012|09}}, 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=September 10, 2012 }}</ref>
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| {| class="wikitable sortable"
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| |+ {{visible anchor|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>
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| |-
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| ! scope="col" |
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| ! scope="col" | ''q''
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| ! scope="col" | ''a''
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| ! scope="col" | ''b''
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| ! scope="col" | ''c''
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| ! scope="col" class="unsortable" | Discovered by
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| |-
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| ! scope="row" | 1
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| | 1.6299 || 2 || 3<sup>10</sup>·109 || 23<sup>5</sup> || Eric Reyssat
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| |-
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| ! scope="row" | 2
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| | 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
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| |-
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| ! scope="row" | 3
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| | 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
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| |-
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| ! scope="row" | 4
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| | 1.5808 || 283 || 5<sup>11</sup>·13<sup>2</sup> || 2<sup>8</sup>·3<sup>8</sup>·17<sup>3</sup> || Jerzy Browkin, Juliusz Brzezinski, Abderrahmane Nitaj
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| |-
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| ! scope="row" | 5
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| | 1.5679 || 1 || 2·3<sup>7</sup> || 5<sup>4</sup>·7 || Benne de Weger
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| |}
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| Note: the ''quality'' ''q''(''a'', ''b'', ''c'') of the triple (''a'', ''b'', ''c'') is defined above.
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| ==Refined forms and generalizations==
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| A stronger inequality proposed by {{Harvtxt|Baker|1998}} states that in the [[inequality (mathematics)|inequality]], one can replace rad(''abc'') by
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| :ε<sup>−ω</sup>rad(''abc'')
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| where ω is the total number of distinct primes dividing ''a'', ''b'' and ''c'' {{Harv|Bombieri|Gubler|2006|p=404}}.
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| {{Harvtxt|Baker|1998}} also describes related conjectures of [[Andrew Granville]] that would give upper bounds on ''c'' of the form
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| :<math> K^{\Omega(a b c)} \mathrm{rad}(a b c) \ </math>
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| where Ω(''n'') is the total number of prime factors of ''n'' and
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| :<math> O(\mathrm{rad}(a b c) \Theta(a b c)) \ </math>
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| where Θ(''n'') is the number of integers up to ''n'' divisible only by primes dividing ''n''.
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| {{harvtxt|Browkin|Brzeziński|1994}} formulated the ''n''-conjecture—a version of the ''abc'' conjecture involving ''n'' > 2 integers.
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| ==Attempts at solution==
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| [[Lucien Szpiro]] attempted a solution in 2007 but it was found to be incorrect.<ref>"Finiteness Theorems for Dynamical Systems", [[Lucien Szpiro]], talk at Conference on L-functions and Automorphic Forms (on the occasion of Dorian Goldfeld's 60th Birthday), Columbia University, May 2007. See {{citation|title=Proof of the abc Conjecture?|first=Peter|last=Woit|authorlink=Peter Woit|work=Not Even Wrong|url=http://www.math.columbia.edu/~woit/wordpress/?p=561|date=May 26, 2007}}.</ref>
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| In August 2012, [[Shinichi Mochizuki]] released a series of four preprints containing a claim to a proof of the ''abc'' conjecture. Mochizuki calls the theory on which this proof is based "[[inter-universal Teichmüller theory]]", and it has other applications including a proof of [[Szpiro's conjecture]] and [[Vojta's conjecture]].<ref name=Mochizukiweb>Mochizuki, Shinichi (August 2012). ''Inter-universal Teichmuller Theory I: Construction of Hodge Theaters'', ''Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic Evaluation'', ''Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice.'', ''Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations'', available at http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html</ref><ref>{{citation|url=http://www.nature.com/news/proof-claimed-for-deep-connection-between-primes-1.11378|title=Proof claimed for deep connection between primes|journal=[[Nature (journal)|Nature]]|date=10 September 2012|first=Phillip|last=Ball|authorlink=Philip Ball}}.</ref> Experts were expected to take months to check Mochizuki's new mathematical machinery, which was developed over decades in 500 pages of preprints and several of his prior papers.<ref>{{citation|title=ABC Proof Could Be Mathematical Jackpot|journal=[[Science (journal)|Science]]|first=Barry|last=Cipra|url=http://news.sciencemag.org/sciencenow/2012/09/abc-conjecture.html|date=September 12, 2012}}.</ref> Attempts at verifying Mochizuki's work are severely hampered by his refusal to leave his home university and lecture on his new mathematics, as is standard in the academy.<ref>[http://projectwordsworth.com/the-paradox-of-the-proof/ The Paradox of the Proof]</ref>
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| When an error in one of the articles was pointed out by [[Vesselin Dimitrov]] and [[Akshay Venkatesh]] in October 2012, Mochizuki posted a comment on his website acknowledging the mistake, stating that it would not affect the result, and promising a corrected version in the near future.<ref>{{cite news|title=An ABC proof too tough even for mathematicians|author=Kevin Hartnett|date=3 November 2012|newspaper=Boston Globe|url=http://www.bostonglobe.com/ideas/2012/11/03/abc-proof-too-tough-even-for-mathematicians/o9bja4kwPuXhDeDb2Ana2K/story.html}}</ref> He revised all of his papers on "[[inter-universal Teichmüller theory]]" of which the latest 2 revisions dated December 2013.<ref name=Mochizukiweb/> Mochizuki has refused all requests for media interviews, but released a [http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTeich%20Verification%20Report%202013-12.pdf progress report] in December 2013.
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| ==See also==
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| *[[List of unsolved problems in mathematics]]
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| ==Notes==
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| {{reflist|colwidth=45em}}
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| ==References==
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| {{Refbegin}}
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| *{{cite book | last=Baker | first=Alan | authorlink=Alan Baker (mathematician) | chapter=Logarithmic forms and the ''abc''-conjecture | editor-last=Győry | editor-first=Kálmán | title=Number theory. Diophantine, computational and algebraic aspects. Proceedings of the international conference, Eger, Hungary, July 29-August 2, 1996 | location=Berlin | publisher=de Gruyter | pages=37–44 | year=1998 | isbn=3-11-015364-5 | zbl=0973.11047 |ref=harv }}
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| *{{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 }}
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| *{{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 }}
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| * {{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 }}
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| *{{Cite journal |first=Andrzej |last=Dąbrowski |title=On the diophantine equation ''x''! + ''A'' = ''y''<sup>2</sup> | journal=Nieuw Archief voor Wiskunde, IV. |volume=14 |pages=321–324 |year=1996 | zbl=0876.11015 | ref=harv }}
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| *{{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 }}
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| *{{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 }}
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| * {{cite book | zbl=1046.11035 | last=Goldfeld | first=Dorian | authorlink=Dorian M. Goldfeld | chapter=Modular forms, elliptic curves and the abc-conjecture | editor-last=Wüstholz | editor-first=Gisbert | editor-link=Gisbert Wüstholz | title=A panorama in number theory or The view from Baker's garden. Based on a conference in honor of Alan Baker's 60th birthday, Zürich, Switzerland, 1999 | location=Cambridge | publisher=[[Cambridge University Press]] | pages=128–147 | year=2002 | isbn=0-521-80799-9 | ref=harv }}
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| * {{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 }}
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| *{{Cite journal |last=Granville |first=A. |authorlink=Andrew Granville |year=1998 |title=ABC Allows Us to Count Squarefrees |url=http://www.dms.umontreal.ca/~andrew/PDF/polysq3.pdf |journal=International Mathematics Research Notices |volume=1998 |pages=991-1009 |doi=10.1155/S1073792898000592 |ref=harv}}
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| *{{cite journal |last1=Granville |first1=Andrew | authorlink=Andrew Granville |last2=Stark |first2=H. |year=2000 | title=ABC implies no "Siegel zeros" for L-functions of characters with negative exponent | url= http://www.dms.umontreal.ca/~andrew/PDF/NoSiegelfinal.pdf |journal=[[Inventiones Mathematicae]] |volume= 139 |pages=509–523 |ref=harv }}
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| *{{cite journal |last1=Granville |first1=Andrew | authorlink=Andrew Granville |last2=Tucker |first2=Thomas |year=2002 | title=It’s As Easy As abc | url= http://www.ams.org/notices/200210/fea-granville.pdf |journal=[[Notices of the AMS]] |volume= 49 | issue=10 |pages=1224–1231 |ref=harv }}
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| * {{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 }}
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| * {{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 }}
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| *{{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}}
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| *{{Cite book | last=Masser | first=D. W. | author-link=David Masser | editor1-last=Chen | editor1-first=W. W. L. | title=Proceedings of the Symposium on Analytic Number Theory | publisher=Imperial College | location=London | year=1985 | chapter=Open problems}}
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| *{{cite journal | last=Mollin | first=R.A. | title=A note on the ABC-conjecture | journal=Far East J. Math. Sci. | volume=33 | number=3 | pages=267–275 | year=2009 | issn=0972-0871 | url=http://people.ucalgary.ca/~ramollin/abcconj.pdf | zbl=1241.11034 }}
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| *{{cite book | zbl=1200.11002 | last=Mollin | first=Richard A. | title=Advanced number theory with applications | location=Boca Raton, FL | publisher=CRC Press | year=2010 | isbn=978-1-4200-8328-6 }}
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| *{{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}}
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| *{{Citation | last1=Oesterlé | first1=Joseph | authorlink=Joseph Oesterlé | title=Nouvelles approches du "théorème" de Fermat | url= http://www.numdam.org/item?id=SB_1987-1988__30__165_0 | series=Séminaire Bourbaki exp 694 | id={{MR|992208|}} | year=1988 | journal=Astérisque | issn=0303-1179 | issue=161 | pages=165–186}}
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| *{{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 }}
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| *{{Cite journal |last=Silverman |first=Joseph H. | authorlink=Joseph H. Silverman | 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 }}
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| *{{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 }}
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| *{{Cite journal |last=Stewart |first=C. L. |authorlink=Cameron Leigh Stewart|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 }}
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| *{{Cite journal |last=Stewart |first=C. L. |authorlink=Cameron Leigh Stewart|first2=Kunrui |last2=Yu |authorlink2=Kunrui 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 }}
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| {{Refend}}
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| ==External links==
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| * [http://abcathome.com/ ABC@home] [[Distributed Computing]] project called [[ABC@Home]].
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| * [http://bit-player.org/2007/easy-as-abc Easy as ABC]: Easy to follow, detailed explanation by Brian Hayes.
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| * {{MathWorld | urlname=abcConjecture | title=abc Conjecture}}
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| * Abderrahmane Nitaj's [http://www.math.unicaen.fr/~nitaj/abc.html ABC conjecture home page]
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| * Bart de Smit's [http://www.math.leidenuniv.nl/~desmit/abc/ ABC Triples webpage]
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| * http://www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf
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| * [http://www.sciencenews.org/sn_arc97/12_6_97/mathland.htm The amazing ABC conjecture]
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| * [http://www.thehcmr.org/issue1_1/elkies.pdf The ABC's of Number Theory] by [[Noam D. Elkies]]
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| * [http://www.math.harvard.edu/~mazur/papers/scanQuest.pdf Questions about Number] by [[Barry Mazur]]
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| * [http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture Philosophy behind Mochizuki’s work on the ABC conjecture] on [[MathOverflow]]
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| * [http://michaelnielsen.org/polymath1/index.php?title=ABC_conjecture ABC Conjecture] [[Polymath project]] wiki page linking to various sources of commentary on Mochizuki's papers.
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| * [http://www.youtube.com/watch?v=RkBl7WKzzRw abc Conjecture] Numberphile video
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| [[Category:Conjectures]]
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| [[Category:Number theory]]
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| [[Category:Unsolved problems in mathematics]]
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