abc conjecture

<h2 id="formulations">Formulations</h2>
Before stating the conjecture, the notion of the <a href="/facts/Radical_of_an_integer/xyoLHVNs">radical of an integer</a> must be introduced: for a <a href="/facts/Positive_integer/u65og8JE">positive integer</a> 
 
 
 
 n
 
 
 {\displaystyle n}
 
, the radical of 
 
 
 
 n
 
 
 {\displaystyle n}
 
, denoted 
 
 
 
 
 rad
 
 (
 n
 )
 
 
 {\displaystyle {\text{rad}}(n)}
 
, is the product of the distinct <a href="/facts/Prime_factor/L20FLkgn">prime factors</a> of 
 
 
 
 n
 
 
 {\displaystyle n}
 
. For example,

 
 
 
 
 rad
 
 (
 16
 )
 =
 
 rad
 
 (
 
 2
 
 4
 
 
 )
 =
 
 rad
 
 (
 2
 )
 =
 2
 
 
 {\displaystyle {\text{rad}}(16)={\text{rad}}(2^{4})={\text{rad}}(2)=2}

rad
 
 (
 17
 )
 =
 17
 
 
 {\displaystyle {\text{rad}}(17)=17}

rad
 
 (
 18
 )
 =
 
 rad
 
 (
 2
 ⋅
 
 3
 
 2
 
 
 )
 =
 2
 ⋅
 3
 =
 6
 
 
 {\displaystyle {\text{rad}}(18)={\text{rad}}(2\cdot 3^{2})=2\cdot 3=6}
 


 
 
 
 
 rad
 
 (
 1000000
 )
 =
 
 rad
 
 (
 
 2
 
 6
 
 
 ⋅
 
 5
 
 6
 
 
 )
 =
 2
 ⋅
 5
 =
 10
 
 
 {\displaystyle {\text{rad}}(1000000)={\text{rad}}(2^{6}\cdot 5^{6})=2\cdot 5=10}
 

If a, b, and c are <a href="/facts/Coprime/bnCxCXxs">coprime</a><a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> positive integers such that a + b = c, it turns out that "usually" 
 
 
 
 c
 <
 
 rad
 
 (
 a
 b
 c
 )
 
 
 {\displaystyle c<{\text{rad}}(abc)}
 
. The abc conjecture deals with the exceptions. Specifically, it states that:

For every positive <a href="/facts/Real_number/R02gw5Pb">real number</a> ε, there exist only finitely many triples (a, b, c) of coprime positive integers, with a + b = c, such that<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>

c
        >
        rad
        ⁡
        (
        a
        b
        c
        
          )
          
            1
            +
            ε
          
        
        .
      
    
    {\displaystyle c>\operatorname {rad} (abc)^{1+\varepsilon }.}

An equivalent formulation is:

For every positive real number ε, there exists a constant Kε such that for all triples (a, b, c) of coprime positive integers, with a + b = c:<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>

c
 <
 
 K
 
 ε
 
 
 ⋅
 rad
 ⁡
 (
 a
 b
 c
 
 )
 
 1
 +
 ε
 
 
 .
 
 
 {\displaystyle c<K_{\varepsilon }\cdot \operatorname {rad} (abc)^{1+\varepsilon }.}

Equivalently (using the <a href="/facts/Little_o_notation/weFFjSWg">little o notation</a>):

For all triples (a, b, c) of coprime positive integers with a + b = c, rad(abc) is at least c1-o(1).
A fourth equivalent formulation of the conjecture involves the quality q(a, b, c) of the triple (a, b, c), which is defined as

q
        (
        a
        ,
        b
        ,
        c
        )
        =
        
          
            
              log
              ⁡
              (
              c
              )
            
            
              log
              ⁡
              
                
                  (
                
              
              
                
                  rad
                
              
              (
              a
              b
              c
              )
              
                
                  )
                
              
            
          
        
        .
      
    
    {\displaystyle q(a,b,c)={\frac {\log(c)}{\log {\big (}{\textrm {rad}}(abc){\big )}}}.}

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 < 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 <a href="/facts/Prime_number/L20FLkgn">prime numbers</a>. The fourth formulation is:

For every positive real number ε, 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. In particular, if the conjecture is true, then there must exist a triple (a, b, c) that achieves the maximal possible quality q(a, b, c).

<h2 id="examples-of-triples-with-small-radical">Examples of triples with small radical</h2>
The condition that ε > 0 is necessary as there exist infinitely many triples a, b, c with c > rad(abc). For example, let

a
        =
        1
        ,
        
        b
        =
        
          2
          
            6
            n
          
        
        −
        1
        ,
        
        c
        =
        
          2
          
            6
            n
          
        
        ,
        
        n
        >
        1.
      
    
    {\displaystyle a=1,\quad b=2^{6n}-1,\quad c=2^{6n},\qquad n>1.}

The integer b is divisible by 9:

b
        =
        
          2
          
            6
            n
          
        
        −
        1
        =
        
          64
          
            n
          
        
        −
        1
        =
        (
        64
        −
        1
        )
        (
        ⋯
        )
        =
        9
        ⋅
        7
        ⋅
        (
        ⋯
        )
        .
      
    
    {\displaystyle b=2^{6n}-1=64^{n}-1=(64-1)(\cdots )=9\cdot 7\cdot (\cdots ).}

Using this fact, the following calculation is made:

rad
 ⁡
 (
 a
 b
 c
 )
 
 
 
 =
 rad
 ⁡
 (
 a
 )
 rad
 ⁡
 (
 b
 )
 rad
 ⁡
 (
 c
 )
 
 
 
 
 
 
 =
 rad
 ⁡
 (
 1
 )
 rad
 ⁡
 
 (
 
 
 2
 
 6
 n
 
 
 −
 1
 
 )
 
 rad
 ⁡
 
 (
 
 2
 
 6
 n
 
 
 )
 
 
 
 
 
 
 
 =
 2
 rad
 ⁡
 
 (
 
 
 2
 
 6
 n
 
 
 −
 1
 
 )
 
 
 
 
 
 
 
 =
 2
 rad
 ⁡
 
 (
 
 9
 ⋅
 
 
 
 b
 9
 
 
 
 
 )
 
 
 
 
 
 
 
 ⩽
 2
 ⋅
 3
 ⋅
 
 
 
 b
 9
 
 
 
 
 
 
 
 
 
 =
 
 
 
 2
 3
 
 
 
 b
 
 
 
 
 
 
 <
 
 
 
 2
 3
 
 
 
 c
 .
 
 
 
 
 
 
 {\displaystyle {\begin{aligned}\operatorname {rad} (abc)&=\operatorname {rad} (a)\operatorname {rad} (b)\operatorname {rad} (c)\\&=\operatorname {rad} (1)\operatorname {rad} \left(2^{6n}-1\right)\operatorname {rad} \left(2^{6n}\right)\\&=2\operatorname {rad} \left(2^{6n}-1\right)\\&=2\operatorname {rad} \left(9\cdot {\tfrac {b}{9}}\right)\\&\leqslant 2\cdot 3\cdot {\tfrac {b}{9}}\\&={\tfrac {2}{3}}b\\&<{\tfrac {2}{3}}c.\end{aligned}}}

By replacing the exponent 6n with other exponents forcing b to have larger square factors, the ratio between the radical and c can be made arbitrarily small. Specifically, let p > 2 be a prime and consider

a
        =
        1
        ,
        
        b
        =
        
          2
          
            p
            (
            p
            −
            1
            )
            n
          
        
        −
        1
        ,
        
        c
        =
        
          2
          
            p
            (
            p
            −
            1
            )
            n
          
        
        ,
        
        n
        >
        1.
      
    
    {\displaystyle a=1,\quad b=2^{p(p-1)n}-1,\quad c=2^{p(p-1)n},\qquad n>1.}

Now it may be plausibly claimed that b is divisible by p2:

b
              
              
                
                =
                
                  2
                  
                    p
                    (
                    p
                    −
                    1
                    )
                    n
                  
                
                −
                1
              
            
            
              
              
                
                =
                
                  
                    (
                    
                      2
                      
                        p
                        (
                        p
                        −
                        1
                        )
                      
                    
                    )
                  
                  
                    n
                  
                
                −
                1
              
            
            
              
              
                
                =
                
                  (
                  
                    
                      2
                      
                        p
                        (
                        p
                        −
                        1
                        )
                      
                    
                    −
                    1
                  
                  )
                
                (
                ⋯
                )
              
            
            
              
              
                
                =
                
                  p
                  
                    2
                  
                
                ⋅
                r
                (
                ⋯
                )
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}b&=2^{p(p-1)n}-1\\&=\left(2^{p(p-1)}\right)^{n}-1\\&=\left(2^{p(p-1)}-1\right)(\cdots )\\&=p^{2}\cdot r(\cdots ).\end{aligned}}}

The last step uses the fact that p2 divides 2p(p−1) − 1. This follows from <a href="/facts/Fermat%27s_little_theorem/qIjSIm2s">Fermat's little theorem</a>, which shows that, for p > 2, 2p−1 = pk + 1 for some integer k. Raising both sides to the power of p then shows that 2p(p−1) = p2(...) + 1.
And now with a similar calculation as above, the following results:

rad
 ⁡
 (
 a
 b
 c
 )
 <
 
 
 
 2
 p
 
 
 
 c
 .
 
 
 {\displaystyle \operatorname {rad} (abc)<{\tfrac {2}{p}}c.}

A list of the highest-quality triples (triples with a particularly small radical relative to c) is given below; the highest quality, 1.6299, was found by Eric Reyssat (Lando & Zvonkin 2004, p. 137) for

a = 2,
b = 310·109 = 6436341,
c = 235 = 6436343,
rad(abc) = 15042.
<h2 id="some-consequences">Some consequences</h2>
The abc conjecture has a large number of consequences. These include both known results (some of which have been proven separately only since the conjecture has been stated) and conjectures for which it gives a <a href="/facts/Conditional_proof/NWrudf6N">conditional proof</a>. The consequences include:

<ul><li><a href="/facts/Roth%27s_theorem/ll7ubGmd">Roth's theorem</a> on <a href="/facts/Diophantine_approximation/WGVMuWF4">Diophantine approximation</a> of <a href="/facts/Algebraic_number/MkH1PzTK">algebraic numbers</a>.[10]<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a></li>
<li>The <a href="/facts/Faltings%27s_theorem/GhyRaknq">Mordell conjecture</a> (already proven in general by <a href="/facts/Gerd_Faltings/9cC0L9WP">Gerd Faltings</a>).<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a></li>
<li>As equivalent, <a href="/facts/Vojta%27s_conjecture/cwmLbggM">Vojta's conjecture</a> in dimension 1.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a></li>
<li>The <a href="/facts/Erd%C5%91s%E2%80%93Woods_number/3D72vQ8x">Erdős–Woods conjecture</a> allowing for a finite number of counterexamples.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a></li>
<li>The existence of infinitely many non-<a href="/facts/Wieferich_prime/tkc913fM">Wieferich primes</a> in every base b > 1.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a></li>
<li>The weak form of <a href="/facts/Marshall_Hall%27s_conjecture/38e462uo">Marshall Hall's conjecture</a> on the separation between squares and cubes of integers.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a></li>
<li><a href="/facts/Fermat%27s_Last_Theorem/fHRc2t8z">Fermat's Last Theorem</a> has <a href="/facts/Wiles%27s_proof_of_Fermat%27s_Last_Theorem/Xgr6wAf2">a famously difficult proof by Andrew Wiles</a>. However it follows easily, at least for 
 
 
 
 n
 ≥
 6
 
 
 {\displaystyle n\geq 6}
 
, from an effective form of a weak version of the abc conjecture. The abc conjecture says the <a href="/facts/Limit_superior_and_limit_inferior/QeAhlygs">lim sup</a> of the set of all qualities (defined above) is 1, which implies the much weaker assertion that there is a finite upper bound for qualities. The conjecture that 2 is such an upper bound suffices for a very short proof of Fermat's Last Theorem for 
 
 
 
 n
 ≥
 6
 
 
 {\displaystyle n\geq 6}
 
.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a></li>
<li>The <a href="/facts/Fermat%E2%80%93Catalan_conjecture/RxRccY0M">Fermat–Catalan conjecture</a>, a generalization of <a href="/facts/Fermat%27s_Last_Theorem/fHRc2t8z">Fermat's Last Theorem</a> concerning powers that are sums of powers.<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a></li>
<li>The <a href="/facts/Dirichlet_L-function/my8hc6Qy">L-function</a> L(s, χd) formed with the <a href="/facts/Legendre_symbol/MwHrJ7as">Legendre symbol</a>, has no <a href="/facts/Siegel_zero/2sSmBoxW">Siegel zero</a>, given a uniform version of the abc conjecture in <a href="/facts/Number_field/e0K0TXSh">number fields</a>, not just the abc conjecture as formulated above for rational integers.<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a></li>
<li>A <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> P(x) has only finitely many <a href="/facts/Perfect_powers/HSzSQBvb">perfect powers</a> for all <a href="/facts/Integers/PUwwrawx">integers</a> x if P has at least three <a href="/facts/Simple_zero/dndUyetA">simple zeros</a>.<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a></li>
<li>A generalization of <a href="/facts/Tijdeman%27s_theorem/SE3masyS">Tijdeman's theorem</a> concerning the number of solutions of ym = xn + k (Tijdeman's theorem answers the case k = 1), and Pillai's conjecture (1931) concerning the number of solutions of Aym = Bxn + k.</li>
<li>As equivalent, 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 that for all coprime integers x, y, the radical of f(x, y) exceeds C · max{|x|, |y|}n−β.<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a></li>
<li>all the polynominals (x^n-1)/(x-1) have an infinity of square-free values.<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a></li>
<li>As equivalent, the modified <a href="/facts/Szpiro_conjecture/LqYE6vK5">Szpiro conjecture</a>, which would yield a bound of rad(abc)1.2+ε.<a class="footnote-ref" id="fnref:25" href="#fn:25">25</a></li>
<li>Dąbrowski (1996) has shown that the abc conjecture implies that <a href="/facts/Brocard%27s_problem/yf224Jby">the Diophantine equation n! + A = k2</a> has only finitely many solutions for any given integer A.</li>
<li>There are ~cfN positive integers n ≤ N for which f(n)/B' is square-free, with cf > 0 a positive constant defined as:<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a>
 
 
 
 
 c
 
 f
 
 
 =
 
 ∏
 
 
 prime 
 
 p
 
 
 
 x
 
 i
 
 
 
 (
 
 1
 −
 
 
 
 ω
 
 
 
 
 f
 
 
 (
 p
 )
 
 
 p
 
 2
 +
 
 q
 
 p
 
 
 
 
 
 
 
 )
 
 .
 
 
 {\displaystyle c_{f}=\prod _{{\text{prime }}p}x_{i}\left(1-{\frac {\omega \,\!_{f}(p)}{p^{2+q_{p}}}}\right).}
 
</li>
<li>The <a href="/facts/Beal_conjecture/cWWrfR5d">Beal conjecture</a>, a generalization of Fermat's Last Theorem proposing that if A, B, C, x, y, and z are positive integers with Ax + By = Cz and x, y, z > 2, then A, B, and C have a common prime factor. The abc conjecture would imply that there are only finitely many counterexamples.</li>
<li><a href="/facts/N%C3%A9ron%E2%80%93Tate_height/IpAkTcqU">Lang's conjecture</a>, a lower bound for the <a href="/facts/Height_function/ByeW10AJ">height</a> of a non-torsion rational point of an elliptic curve.</li>
<li>A negative solution to the <a href="/facts/Erd%C5%91s%E2%80%93Ulam_problem/ptXjS5qt">Erdős–Ulam problem</a> on dense sets of Euclidean points with rational distances.<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a></li>
<li>An effective version of <a href="/facts/Siegel%27s_theorem_on_integral_points/J2AGT4ew">Siegel's theorem about integral points on algebraic curves</a>.<a class="footnote-ref" id="fnref:28" href="#fn:28">28</a></li></ul>
<h2 id="theoretical-results">Theoretical results</h2>
The abc conjecture implies that c can be <a href="/facts/Upper_bound/YJje9oC2">bounded above</a> by a near-linear function of the radical of abc. Bounds are known that are <a href="/facts/Exponential_function/ewlYxvR2">exponential</a>. Specifically, the following bounds have been proven:

c
 <
 exp
 ⁡
 
 
 (
 
 
 K
 
 1
 
 
 rad
 ⁡
 (
 a
 b
 c
 
 )
 
 15
 
 
 
 )
 
 
 
 
 {\displaystyle c<\exp {\left(K_{1}\operatorname {rad} (abc)^{15}\right)}}
 
 (Stewart & Tijdeman 1986),

c
 <
 exp
 ⁡
 
 
 (
 
 
 K
 
 2
 
 
 rad
 ⁡
 (
 a
 b
 c
 
 )
 
 
 
 2
 3
 
 
 +
 ε
 
 
 
 )
 
 
 
 
 {\displaystyle c<\exp {\left(K_{2}\operatorname {rad} (abc)^{{\frac {2}{3}}+\varepsilon }\right)}}
 
 (Stewart & Yu 1991), and

c
 <
 exp
 ⁡
 
 
 (
 
 
 K
 
 3
 
 
 rad
 ⁡
 (
 a
 b
 c
 
 )
 
 
 1
 3
 
 
 
 
 
 (
 
 log
 ⁡
 (
 rad
 ⁡
 (
 a
 b
 c
 )
 
 )
 
 
 3
 
 
 
 )
 
 
 
 
 {\displaystyle c<\exp {\left(K_{3}\operatorname {rad} (abc)^{\frac {1}{3}}\left(\log(\operatorname {rad} (abc)\right)^{3}\right)}}
 
 (Stewart & Yu 2001).
In these bounds, K1 and K3 are <a href="/facts/Constant_(mathematics)/SxlxTWFT">constants</a> that do not depend on a, b, or c, and K2 is a constant that depends on ε (in an <a href="/facts/Effectively_computable/dzwBlLGn">effectively computable</a> way) but not on a, b, or c. The bounds apply to any triple for which c > 2.
There are also theoretical results that provide a lower bound on the best possible form of the abc conjecture. In particular, Stewart & Tijdeman (1986) showed that there are infinitely many triples (a, b, c) of coprime integers with a + b = c and

c
        >
        rad
        ⁡
        (
        a
        b
        c
        )
        exp
        ⁡
        
          
            (
            
              k
              
                
                  log
                  ⁡
                  c
                
              
              
                /
              
              log
              ⁡
              log
              ⁡
              c
            
            )
          
        
      
    
    {\displaystyle c>\operatorname {rad} (abc)\exp {\left(k{\sqrt {\log c}}/\log \log c\right)}}

for all k < 4. The constant k was improved to k = 6.068 by van Frankenhuysen (2000).

<h2 id="computational-results">Computational results</h2>
In 2006, the Mathematics Department of <a href="/facts/Leiden_University/MqnmnTtF">Leiden University</a> in the Netherlands, together with the Dutch Kennislink science institute, launched the <a href="/facts/ABC@Home/nBLT1c1g">ABC@Home</a> project, a <a href="/facts/Grid_computing/2GQhzN5l">grid computing</a> 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.

Distribution of triples with q > 1<a class="footnote-ref" id="fnref:29" href="#fn:29">29</a><table><tbody><tr><th scope="col">qc</th><th scope="col">q > 1</th><th scope="col">q > 1.05</th><th scope="col">q > 1.1</th><th scope="col">q > 1.2</th><th scope="col">q > 1.3</th><th scope="col">q > 1.4</th></tr><tr><th scope="row">c < 102</th><td>6</td><td>4</td><td>4</td><td>2</td><td>0</td><td>0</td></tr><tr><th scope="row">c < 103</th><td>31</td><td>17</td><td>14</td><td>8</td><td>3</td><td>1</td></tr><tr><th scope="row">c < 104</th><td>120</td><td>74</td><td>50</td><td>22</td><td>8</td><td>3</td></tr><tr><th scope="row">c < 105</th><td>418</td><td>240</td><td>152</td><td>51</td><td>13</td><td>6</td></tr><tr><th scope="row">c < 106</th><td>1,268</td><td>667</td><td>379</td><td>102</td><td>29</td><td>11</td></tr><tr><th scope="row">c < 107</th><td>3,499</td><td>1,669</td><td>856</td><td>210</td><td>60</td><td>17</td></tr><tr><th scope="row">c < 108</th><td>8,987</td><td>3,869</td><td>1,801</td><td>384</td><td>98</td><td>25</td></tr><tr><th scope="row">c < 109</th><td>22,316</td><td>8,742</td><td>3,693</td><td>706</td><td>144</td><td>34</td></tr><tr><th scope="row">c < 1010</th><td>51,677</td><td>18,233</td><td>7,035</td><td>1,159</td><td>218</td><td>51</td></tr><tr><th scope="row">c < 1011</th><td>116,978</td><td>37,612</td><td>13,266</td><td>1,947</td><td>327</td><td>64</td></tr><tr><th scope="row">c < 1012</th><td>252,856</td><td>73,714</td><td>23,773</td><td>3,028</td><td>455</td><td>74</td></tr><tr><th scope="row">c < 1013</th><td>528,275</td><td>139,762</td><td>41,438</td><td>4,519</td><td>599</td><td>84</td></tr><tr><th scope="row">c < 1014</th><td>1,075,319</td><td>258,168</td><td>70,047</td><td>6,665</td><td>769</td><td>98</td></tr><tr><th scope="row">c < 1015</th><td>2,131,671</td><td>463,446</td><td>115,041</td><td>9,497</td><td>998</td><td>112</td></tr><tr><th scope="row">c < 1016</th><td>4,119,410</td><td>812,499</td><td>184,727</td><td>13,118</td><td>1,232</td><td>126</td></tr><tr><th scope="row">c < 1017</th><td>7,801,334</td><td>1,396,909</td><td>290,965</td><td>17,890</td><td>1,530</td><td>143</td></tr><tr><th scope="row">c < 1018</th><td>14,482,065</td><td>2,352,105</td><td>449,194</td><td>24,013</td><td>1,843</td><td>160</td></tr></tbody></table>
As of May 2014, <a href="/facts/ABC@Home/nBLT1c1g">ABC@Home</a> had found 23.8 million triples.<a class="footnote-ref" id="fnref:30" href="#fn:30">30</a>

Highest-quality triples<a class="footnote-ref" id="fnref:31" href="#fn:31">31</a><table><tbody><tr><th scope="col">Rank</th><th scope="col">q</th><th scope="col">a</th><th scope="col">b</th><th scope="col">c</th><th scope="col">Discovered by</th></tr><tr><th scope="row">1</th><td>1.6299</td><td>2</td><td>310·109</td><td>235</td><td>Eric Reyssat</td></tr><tr><th scope="row">2</th><td>1.6260</td><td>112</td><td>32·56·73</td><td>221·23</td><td>Benne de Weger</td></tr><tr><th scope="row">3</th><td>1.6235</td><td>19·1307</td><td>7·292·318</td><td>28·322·54</td><td>Jerzy Browkin, Juliusz Brzezinski</td></tr><tr><th scope="row">4</th><td>1.5808</td><td>283</td><td>511·132</td><td>28·38·173</td><td>Jerzy Browkin, Juliusz Brzezinski, Abderrahmane Nitaj</td></tr><tr><th scope="row">5</th><td>1.5679</td><td>1</td><td>2·37</td><td>54·7</td><td>Benne de Weger</td></tr></tbody></table>
Note: the quality q(a, b, c) of the triple (a, b, c) is defined above.

<h2 id="refined-forms-generalizations-and-related-statements">Refined forms, generalizations and related statements</h2>
The abc conjecture is an integer analogue of the <a href="/facts/Mason%E2%80%93Stothers_theorem/YxzKGCAk">Mason–Stothers theorem</a> for polynomials.
A strengthening, proposed by Baker (1998), states that in the abc conjecture one can replace rad(abc) by

ε−ω rad(abc),
where ω is the total number of distinct primes dividing a, b and c.<a class="footnote-ref" id="fnref:32" href="#fn:32">32</a>
<a href="/facts/Andrew_Granville/EhFz6kyI">Andrew Granville</a> noticed that the minimum of the function 
 
 
 
 
 
 (
 
 
 
 ε
 
 −
 ω
 
 
 rad
 ⁡
 (
 a
 b
 c
 )
 
 
 
 )
 
 
 
 1
 +
 ε
 
 
 
 
 {\displaystyle {\big (}\varepsilon ^{-\omega }\operatorname {rad} (abc){\big )}^{1+\varepsilon }}
 
 over 
 
 
 
 ε
 >
 0
 
 
 {\displaystyle \varepsilon >0}
 
 occurs when 
 
 
 
 ε
 =
 
 
 ω
 
 log
 ⁡
 
 
 (
 
 
 rad
 ⁡
 (
 a
 b
 c
 )
 
 
 )
 
 
 
 
 
 .
 
 
 {\displaystyle \varepsilon ={\frac {\omega }{\log {\big (}\operatorname {rad} (abc){\big )}}}.}

This inspired Baker (2004) to propose a sharper form of the abc conjecture, namely:

c
 <
 κ
 rad
 ⁡
 (
 a
 b
 c
 )
 
 
 
 
 
 (
 
 
 log
 ⁡
 
 
 (
 
 
 rad
 ⁡
 (
 a
 b
 c
 )
 
 
 )
 
 
 
 
 
 )
 
 
 
 ω
 
 
 
 
 ω
 !
 
 
 
 
 
 {\displaystyle c<\kappa \operatorname {rad} (abc){\frac {{\Big (}\log {\big (}\operatorname {rad} (abc){\big )}{\Big )}^{\omega }}{\omega !}}}

with κ an absolute constant. After some computational experiments he found that a value of 
 
 
 
 6
 
 /
 
 5
 
 
 {\displaystyle 6/5}
 
 was admissible for κ. This version is called the "explicit abc conjecture".
Baker (1998) also describes related conjectures of <a href="/facts/Andrew_Granville/EhFz6kyI">Andrew Granville</a> that would give upper bounds on c of the form

K
          
            Ω
            (
            a
            b
            c
            )
          
        
        rad
        ⁡
        (
        a
        b
        c
        )
        ,
      
    
    {\displaystyle K^{\Omega (abc)}\operatorname {rad} (abc),}

where Ω(n) is the total number of prime factors of n, and

O
        
          
            (
          
        
        rad
        ⁡
        (
        a
        b
        c
        )
        Θ
        (
        a
        b
        c
        )
        
          
            )
          
        
        ,
      
    
    {\displaystyle O{\big (}\operatorname {rad} (abc)\Theta (abc){\big )},}

where Θ(n) is the number of integers up to n divisible only by primes dividing n.
Robert, Stewart & Tenenbaum (2014) proposed a more precise inequality based on Robert & Tenenbaum (2013).
Let k = rad(abc). They conjectured there is a constant C1 such that

c
 <
 k
 exp
 ⁡
 
 (
 
 4
 
 
 
 
 3
 log
 ⁡
 k
 
 
 log
 ⁡
 log
 ⁡
 k
 
 
 
 
 
 (
 
 1
 +
 
 
 
 log
 ⁡
 log
 ⁡
 log
 ⁡
 k
 
 
 2
 log
 ⁡
 log
 ⁡
 k
 
 
 
 +
 
 
 
 C
 
 1
 
 
 
 log
 ⁡
 log
 ⁡
 k
 
 
 
 
 )
 
 
 )
 
 
 
 {\displaystyle c<k\exp \left(4{\sqrt {\frac {3\log k}{\log \log k}}}\left(1+{\frac {\log \log \log k}{2\log \log k}}+{\frac {C_{1}}{\log \log k}}\right)\right)}

holds whereas there is a constant C2 such that

c
        >
        k
        exp
        ⁡
        
          (
          
            4
            
              
                
                  
                    3
                    log
                    ⁡
                    k
                  
                  
                    log
                    ⁡
                    log
                    ⁡
                    k
                  
                
              
            
            
              (
              
                1
                +
                
                  
                    
                      log
                      ⁡
                      log
                      ⁡
                      log
                      ⁡
                      k
                    
                    
                      2
                      log
                      ⁡
                      log
                      ⁡
                      k
                    
                  
                
                +
                
                  
                    
                      C
                      
                        2
                      
                    
                    
                      log
                      ⁡
                      log
                      ⁡
                      k
                    
                  
                
              
              )
            
          
          )
        
      
    
    {\displaystyle c>k\exp \left(4{\sqrt {\frac {3\log k}{\log \log k}}}\left(1+{\frac {\log \log \log k}{2\log \log k}}+{\frac {C_{2}}{\log \log k}}\right)\right)}

holds infinitely often.
Browkin & Brzeziński (1994) formulated the <a href="/facts/N_conjecture/B9Q264fh">n conjecture</a>—a version of the abc conjecture involving n > 2 integers.

<h2 id="claimed-proofs">Claimed proofs</h2>
<a href="/facts/Lucien_Szpiro/LGt2apol">Lucien Szpiro</a> proposed a solution in 2007, but it was found to be incorrect shortly afterwards.<a class="footnote-ref" id="fnref:33" href="#fn:33">33</a>
Since August 2012, <a href="/facts/Shinichi_Mochizuki/24KRK3Gj">Shinichi Mochizuki</a> has claimed a proof of Szpiro's conjecture and therefore the abc conjecture.<a class="footnote-ref" id="fnref:34" href="#fn:34">34</a> He released a series of four preprints developing a new theory he called <a href="/facts/Inter-universal_Teichm%C3%BCller_theory/IklqBJkx">inter-universal Teichmüller theory</a> (IUTT), which is then applied to prove the abc conjecture.<a class="footnote-ref" id="fnref:35" href="#fn:35">35</a>
The papers have not been widely accepted by the mathematical community as providing a proof of abc.<a class="footnote-ref" id="fnref:36" href="#fn:36">36</a> This is not only because of their length and the difficulty of understanding them,<a class="footnote-ref" id="fnref:37" href="#fn:37">37</a> but also because at least one specific point in the argument has been identified as a gap by some other experts.<a class="footnote-ref" id="fnref:38" href="#fn:38">38</a> Although a few mathematicians have vouched for the correctness of the proof<a class="footnote-ref" id="fnref:39" href="#fn:39">39</a> and have attempted to communicate their understanding via workshops on IUTT, they have failed to convince the number theory community at large.<a class="footnote-ref" id="fnref:40" href="#fn:40">40</a><a class="footnote-ref" id="fnref:41" href="#fn:41">41</a>
In March 2018, <a href="/facts/Peter_Scholze/zLFj1Dhl">Peter Scholze</a> and <a href="/facts/Jakob_Stix/6VTXNp90">Jakob Stix</a> visited <a href="/facts/Kyoto_University/fxhusvuc">Kyoto</a> for discussions with Mochizuki.<a class="footnote-ref" id="fnref:42" href="#fn:42">42</a><a class="footnote-ref" id="fnref:43" href="#fn:43">43</a>
While they did not resolve the differences, they brought them into clearer focus.
Scholze and Stix wrote a report asserting and explaining an error in the logic of the proof and claiming that the resulting gap was "so severe that ... small modifications will not rescue the proof strategy";<a class="footnote-ref" id="fnref:44" href="#fn:44">44</a>
Mochizuki claimed that they misunderstood vital aspects of the theory and made invalid simplifications.<a class="footnote-ref" id="fnref:45" href="#fn:45">45</a><a class="footnote-ref" id="fnref:46" href="#fn:46">46</a><a class="footnote-ref" id="fnref:47" href="#fn:47">47</a>
On April 3, 2020, two mathematicians from the Kyoto <a href="/facts/Research_Institute_for_Mathematical_Sciences/5zm9UjfP">research institute</a> where Mochizuki works announced that his claimed proof would be published in Publications of the Research Institute for Mathematical Sciences, the institute's journal. Mochizuki is chief editor of the journal but recused himself from the review of the paper.<a class="footnote-ref" id="fnref:48" href="#fn:48">48</a> The announcement was received with skepticism by <a href="/facts/Kiran_Kedlaya/kzQrFP2E">Kiran Kedlaya</a> and <a href="/facts/Edward_Frenkel/HDbzeeM5">Edward Frenkel</a>, as well as being described by <a href="/facts/Nature_(journal)/XmlJXkFx">Nature</a> as "unlikely to move many researchers over to Mochizuki's camp".<a class="footnote-ref" id="fnref:49" href="#fn:49">49</a> In March 2021, Mochizuki's proof was published in RIMS.<a class="footnote-ref" id="fnref:50" href="#fn:50">50</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/List_of_unsolved_problems_in_mathematics/RroxGPpP">List of unsolved problems in mathematics</a></li></ul>
<h2 id="notes">Notes</h2>

<h2 id="sources">Sources</h2>

<ul><li><a href="/facts/Alan_Baker_(mathematician)/1BAzUSML">Baker, Alan</a> (1998). "Logarithmic forms and the abc-conjecture". In Győry, Kálmán (ed.). Number theory. Diophantine, computational and algebraic aspects. Proceedings of the international conference, Eger, Hungary, July 29-August 2, 1996. Berlin: de Gruyter. pp. 37–44. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-11-015364-5. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0973.11047">0973.11047</a>.</li>
<li><a href="/facts/Alan_Baker_(mathematician)/1BAzUSML">Baker, Alan</a> (2004). <a href="https://publi.math.unideb.hu/load_pdf.php?p=972">"Experiments on the abc-conjecture"</a>. Publicationes Mathematicae Debrecen. 65 (3–4): 253–260. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.5486%2FPMD.2004.3348">10.5486/PMD.2004.3348</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:253834357">253834357</a>.</li>
<li>Bombieri, Enrico (1994). "Roth's theorem and the abc-conjecture" (Preprint). ETH Zürich.[unreliable source?]</li>
<li><a href="/facts/Enrico_Bombieri/MU6xU5vr">Bombieri, Enrico</a>; Gubler, Walter (2006). Heights in Diophantine Geometry. New Mathematical Monographs. Vol. 4. <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-71229-3. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1130.11034">1130.11034</a>.</li>
<li><a href="/facts/Jerzy_Browkin/JQh5G7LL">Browkin, Jerzy</a>; Brzeziński, Juliusz (1994). "Some remarks on the abc-conjecture". Math. Comp. 62 (206): 931–939. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1994MaCom..62..931B">1994MaCom..62..931B</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2153551">10.2307/2153551</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2153551">2153551</a>.</li>
<li>Browkin, Jerzy (2000). "The abc-conjecture". In Bambah, R. P.; Dumir, V. C.; Hans-Gill, R. J. (eds.). <a href="https://archive.org/details/numbertheory00bamb_636">Number Theory</a>. Trends in Mathematics. Basel: Birkhäuser. pp. <a href="https://archive.org/details/numbertheory00bamb_636/page/n83">75</a>–106. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-7643-6259-6.</li>
<li>Dąbrowski, Andrzej (1996). "On the diophantine equation x! + A = y2". Nieuw Archief voor Wiskunde, IV. 14: 321–324. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0876.11015">0876.11015</a>.</li>
<li><a href="/facts/Noam_Elkies/t1Uerz9E">Elkies, N. D.</a> (1991). <a href="https://doi.org/10.1155%2FS1073792891000144">"ABC implies Mordell"</a>. International Mathematics Research Notices. 1991 (7): 99–109. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1155%2FS1073792891000144">10.1155/S1073792891000144</a>.</li>
<li><a href="/facts/Gerhard_Frey/MBQcRzPm">Frey, Gerhard</a> (1997). <a href="https://books.google.com/books?id=Va-quzVwtMsC&pg=PA527">"On Ternary Equations of Fermat Type and Relations with Elliptic Curves"</a>. Modular Forms and Fermat's Last Theorem. New York: Springer. pp. 527–548. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-94609-8.</li>
<li><a href="/facts/Dorian_M._Goldfeld/I9dBLEA6">Goldfeld, Dorian</a> (1996). "Beyond the last theorem". <a href="/facts/Math_Horizons/U9MTEPTm">Math Horizons</a>. 4 (September): 26–34. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1080%2F10724117.1996.11974985">10.1080/10724117.1996.11974985</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/25678079">25678079</a>.</li>
<li><a href="/facts/Dorian_M._Goldfeld/I9dBLEA6">Goldfeld, Dorian</a> (2002). "Modular forms, elliptic curves and the abc-conjecture". In <a href="/facts/Gisbert_W%C3%BCstholz/FSPgk0lN">Wüstholz, Gisbert</a> (ed.). 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. Cambridge: <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. pp. 128–147. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-80799-9. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1046.11035">1046.11035</a>.</li>
<li><a href="/facts/Timothy_Gowers/El3xoEmE">Gowers, Timothy</a>; Barrow-Green, June; Leader, Imre, eds. (2008). <a href="/facts/The_Princeton_Companion_to_Mathematics/MzrNYmAz">The Princeton Companion to Mathematics</a>. Princeton: Princeton University Press. pp. <a href="https://archive.org/details/princetoncompani00gowe_360/page/n672">361</a>–362, 681. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-691-11880-2.</li>
<li><a href="/facts/Andrew_Granville/EhFz6kyI">Granville, A.</a> (1998). <a href="http://www.dms.umontreal.ca/~andrew/PDF/polysq3.pdf">"ABC Allows Us to Count Squarefrees"</a> (PDF). International Mathematics Research Notices. 1998 (19): 991–1009. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1155%2FS1073792898000592">10.1155/S1073792898000592</a>.</li>
<li><a href="/facts/Andrew_Granville/EhFz6kyI">Granville, Andrew</a>; Stark, H. (2000). <a href="http://www.dms.umontreal.ca/~andrew/PDF/NoSiegelfinal.pdf">"ABC implies no "Siegel zeros" for L-functions of characters with negative exponent"</a> (PDF). <a href="/facts/Inventiones_Mathematicae/bARcL95Y">Inventiones Mathematicae</a>. 139 (3): 509–523. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2000InMat.139..509G">2000InMat.139..509G</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs002229900036">10.1007/s002229900036</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:6901166">6901166</a>.</li>
<li><a href="/facts/Andrew_Granville/EhFz6kyI">Granville, Andrew</a>; Tucker, Thomas (2002). <a href="https://www.ams.org/journals/notices/200210/fea-granville.pdf">"It's As Easy As abc"</a> (PDF). <a href="/facts/Notices_of_the_AMS/smHYV3sv">Notices of the AMS</a>. 49 (10): 1224–1231. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.146.610">10.1.1.146.610</a>.</li>
<li><a href="/facts/Richard_K._Guy/shtfXH4N">Guy, Richard K.</a> (2004). Unsolved Problems in Number Theory. Berlin: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-20860-7.</li>
<li>Lando, Sergei K.; Zvonkin, Alexander K. (2004). "Graphs on Surfaces and Their Applications". Encyclopaedia of Mathematical Sciences: Lower-Dimensional Topology II. Vol. 141. Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-00203-0.</li>
<li>Langevin, M. (1993). "Cas d'égalité pour le théorème de Mason et applications de la conjecture abc". Comptes rendus de l'Académie des sciences (in French). 317 (5): 441–444.</li>
<li><a href="/facts/David_Masser/91TBlV3n">Masser, D. W.</a> (1985). "Open problems". In Chen, W. W. L. (ed.). Proceedings of the Symposium on Analytic Number Theory. London: Imperial College.</li>
<li>Mollin, R.A. (2009). <a href="https://web.archive.org/web/20160304053930/http://people.ucalgary.ca/~ramollin/abcconj.pdf">"A note on the ABC-conjecture"</a> (PDF). Far East Journal of Mathematical Sciences. 33 (3): 267–275. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0972-0871">0972-0871</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1241.11034">1241.11034</a>. Archived from <a href="http://people.ucalgary.ca/~ramollin/abcconj.pdf">the original</a> (PDF) on 2016-03-04. Retrieved 2013-06-14.</li>
<li>Mollin, Richard A. (2010). Advanced number theory with applications. Boca Raton, FL: CRC Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4200-8328-6. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1200.11002">1200.11002</a>.</li>
<li>Nitaj, Abderrahmane (1996). "La conjecture abc". Enseign. Math. (in French). 42 (1–2): 3–24.</li>
<li><a href="/facts/Joseph_Oesterl%C3%A9/YBLCcxLA">Oesterlé, Joseph</a> (1988), <a href="http://www.numdam.org/item?id=SB_1987-1988__30__165_0">"Nouvelles approches du "théorème" de Fermat"</a>, Astérisque, Séminaire Bourbaki exp 694 (161): 165–186, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0303-1179">0303-1179</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0992208">0992208</a></li>
<li><a href="/facts/Carl_Pomerance/mmKNsTUr">Pomerance, Carl</a> (2008). "Computational Number Theory". The Princeton Companion to Mathematics. Princeton University Press. pp. 361–362.</li>
<li><a href="/facts/Joseph_H._Silverman/acyPld7K">Silverman, Joseph H.</a> (1988). <a href="https://doi.org/10.1016%2F0022-314X%2888%2990019-4">"Wieferich's criterion and the abc-conjecture"</a>. <a href="/facts/Journal_of_Number_Theory/5CecYxqW">Journal of Number Theory</a>. 30 (2): 226–237. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0022-314X%2888%2990019-4">10.1016/0022-314X(88)90019-4</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0654.10019">0654.10019</a>.</li>
<li>Robert, Olivier; <a href="/facts/Cameron_Leigh_Stewart/5eEsc5Ce">Stewart, Cameron L.</a>; <a href="/facts/G%C3%A9rald_Tenenbaum/pytj23Hm">Tenenbaum, Gérald</a> (2014). <a href="https://hal.archives-ouvertes.fr/hal-01281526/file/abc.pdf">"A refinement of the abc conjecture"</a> (PDF). <a href="/facts/Bulletin_of_the_London_Mathematical_Society/jdcf3vhX">Bulletin of the London Mathematical Society</a>. 46 (6): 1156–1166. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1112%2Fblms%2Fbdu069">10.1112/blms/bdu069</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:123460044">123460044</a>.</li>
<li>Robert, Olivier; Tenenbaum, Gérald (November 2013). <a href="https://doi.org/10.1016%2Fj.indag.2013.07.007">"Sur la répartition du noyau d'un entier"</a> [On the distribution of the kernel of an integer]. Indagationes Mathematicae (in French). 24 (4): 802–914. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2Fj.indag.2013.07.007">10.1016/j.indag.2013.07.007</a>.</li>
<li><a href="/facts/Cameron_Leigh_Stewart/5eEsc5Ce">Stewart, C. L.</a>; <a href="/facts/Robert_Tijdeman/HRI1kHqN">Tijdeman, R.</a> (1986). "On the Oesterlé-Masser conjecture". Monatshefte für Mathematik. 102 (3): 251–257. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01294603">10.1007/BF01294603</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:123621917">123621917</a>.</li>
<li><a href="/facts/Cameron_Leigh_Stewart/5eEsc5Ce">Stewart, C. L.</a>; Yu, Kunrui (1991). "On the abc conjecture". <a href="/facts/Mathematische_Annalen/GPhGaonc">Mathematische Annalen</a>. 291 (1): 225–230. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01445201">10.1007/BF01445201</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:123894587">123894587</a>.</li>
<li><a href="/facts/Cameron_Leigh_Stewart/5eEsc5Ce">Stewart, C. L.</a>; Yu, Kunrui (2001). "On the abc conjecture, II". <a href="/facts/Duke_Mathematical_Journal/S6wn8HYp">Duke Mathematical Journal</a>. 108 (1): 169–181. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1215%2FS0012-7094-01-10815-6">10.1215/S0012-7094-01-10815-6</a>.</li>
<li>van Frankenhuysen, Machiel (2000). <a href="https://doi.org/10.1006%2Fjnth.1999.2484">"A Lower Bound in the abc Conjecture"</a>. <a href="/facts/J._Number_Theory/5CecYxqW">J. Number Theory</a>. 82 (1): 91–95. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1006%2Fjnth.1999.2484">10.1006/jnth.1999.2484</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1755155">1755155</a>.</li>
<li>Van Frankenhuijsen, Machiel (2002). <a href="https://doi.org/10.1006%2Fjnth.2001.2769">"The ABC conjecture implies Vojta's height inequality for curves"</a>. <a href="/facts/J._Number_Theory/5CecYxqW">J. Number Theory</a>. 95 (2): 289–302. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1006%2Fjnth.2001.2769">10.1006/jnth.2001.2769</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1924103">1924103</a>.</li>
<li>Waldschmidt, Michel (2015). <a href="https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/abcLahoreProceedings.pdf">"Lecture on the abc Conjecture and Some of Its Consequences"</a> (PDF). Mathematics in the 21st Century. Springer Proceedings in Mathematics & Statistics. Vol. 98. pp. 211–230. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-0348-0859-0_13">10.1007/978-3-0348-0859-0_13</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-0348-0858-3.</li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="https://web.archive.org/web/20100917032533/http://abcathome.com/apps.php">ABC@home</a> <a href="/facts/Distributed_computing/o5nyhqTF">Distributed computing</a> project called <a href="/facts/ABC@Home/nBLT1c1g">ABC@Home</a>.</li>
<li><a href="http://bit-player.org/2007/easy-as-abc">Easy as ABC</a>: Easy to follow, detailed explanation by Brian Hayes.</li>
<li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/abcConjecture.html">"abc Conjecture"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li>
<li>Abderrahmane Nitaj's <a href="https://nitaj.users.lmno.cnrs.fr/abc.html">ABC conjecture home page</a></li>
<li>Bart de Smit's <a href="http://www.math.leidenuniv.nl/~desmit/abc/">ABC Triples webpage</a></li>
<li><a href="http://www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf">http://www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf</a></li>
<li><a href="http://dash.harvard.edu/bitstream/handle/1/2793857/Elkies%20-%20ABCs%20of%20Number%20Theory.pdf?sequence=2">The ABC's of Number Theory</a> by <a href="/facts/Noam_D._Elkies/t1Uerz9E">Noam D. Elkies</a></li>
<li><a href="http://www.math.harvard.edu/~mazur/papers/scanQuest.pdf">Questions about Number</a> by <a href="/facts/Barry_Mazur/4tGbAnCX">Barry Mazur</a></li>
<li><a href="https://mathoverflow.net/q/106560">Philosophy behind Mochizuki’s work on the ABC conjecture</a> on <a href="/facts/MathOverflow/0XqLBHA5">MathOverflow</a></li>
<li><a href="http://michaelnielsen.org/polymath1/index.php?title=ABC_conjecture">ABC Conjecture</a> <a href="/facts/Polymath_project/LGj7RXQ9">Polymath project</a> wiki page linking to various sources of commentary on Mochizuki's papers.</li>
<li><a href="https://www.youtube.com/watch?v=RkBl7WKzzRw">abc Conjecture</a> Numberphile video</li>
<li><a href="http://www.kurims.kyoto-u.ac.jp/~motizuki/news-english.html">News about IUT by Mochizuki</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Oesterlé 1988. - Oesterlé, Joseph (1988), "Nouvelles approches du "théorème" de Fermat", Astérisque, Séminaire Bourbaki exp 694 (161): 165–186, ISSN 0303-1179, MR 0992208 <a href="http://www.numdam.org/item?id=SB_1987-1988__30__165_0" target="_blank">http://www.numdam.org/item?id=SB_1987-1988__30__165_0</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Masser 1985. - Masser, D. W. (1985). "Open problems". In Chen, W. W. L. (ed.). Proceedings of the Symposium on Analytic Number Theory. London: Imperial College. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Goldfeld 1996. - Goldfeld, Dorian (1996). "Beyond the last theorem". Math Horizons. 4 (September): 26–34. doi:10.1080/10724117.1996.11974985. JSTOR 25678079. <a href="https://doi.org/10.1080%2F10724117.1996.11974985" target="_blank">https://doi.org/10.1080%2F10724117.1996.11974985</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Fesenko, Ivan (September 2015). "Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta-functions, notes on the work of Shinichi Mochizuki". European Journal of Mathematics. 1 (3): 405–440. doi:10.1007/s40879-015-0066-0. <a href="https://doi.org/10.1007%2Fs40879-015-0066-0" target="_blank">https://doi.org/10.1007%2Fs40879-015-0066-0</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Oesterlé 1988. - Oesterlé, Joseph (1988), "Nouvelles approches du "théorème" de Fermat", Astérisque, Séminaire Bourbaki exp 694 (161): 165–186, ISSN 0303-1179, MR 0992208 <a href="http://www.numdam.org/item?id=SB_1987-1988__30__165_0" target="_blank">http://www.numdam.org/item?id=SB_1987-1988__30__165_0</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Ball, Peter (10 September 2012). "Proof claimed for deep connection between primes". Nature. doi:10.1038/nature.2012.11378. Retrieved 19 March 2018. <a href="https://www.nature.com/news/proof-claimed-for-deep-connection-between-primes-1.11378" target="_blank">https://www.nature.com/news/proof-claimed-for-deep-connection-between-primes-1.11378</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Castelvecchi, Davide (9 April 2020). "Mathematical proof that rocked number theory will be published". Nature. 580 (7802): 177. Bibcode:2020Natur.580..177C. doi:10.1038/d41586-020-00998-2. PMID 32246118. S2CID 214786566. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Further comment by P. Scholze at Not Even Wrong math.columbia.edu[self-published source?] <a href="https://www.math.columbia.edu/~woit/wordpress/?p=11709&cpage=1#comment-235940" target="_blank">https://www.math.columbia.edu/~woit/wordpress/?p=11709&cpage=1#comment-235940</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Scholze, Peter. "Review of inter-universal Teichmüller Theory I". zbmath open. Retrieved 2025-02-25. <a href="https://zbmath.org/1465.14002" target="_blank">https://zbmath.org/1465.14002</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">When a + b = c, any common factor of two of the values is necessarily shared by the third. Thus, coprimality of a, b, c implies pairwise coprimality of a, b, c. So in this case, it does not matter which concept we use. <a href="/wiki/Pairwise_coprime" target="_blank">/wiki/Pairwise_coprime</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Waldschmidt 2015. - Waldschmidt, Michel (2015). "Lecture on the abc Conjecture and Some of Its Consequences" (PDF). Mathematics in the 21st Century. Springer Proceedings in Mathematics & Statistics. Vol. 98. pp. 211–230. doi:10.1007/978-3-0348-0859-0_13. ISBN 978-3-0348-0858-3. <a href="https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/abcLahoreProceedings.pdf" target="_blank">https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/abcLahoreProceedings.pdf</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Waldschmidt 2015. - Waldschmidt, Michel (2015). "Lecture on the abc Conjecture and Some of Its Consequences" (PDF). Mathematics in the 21st Century. Springer Proceedings in Mathematics & Statistics. Vol. 98. pp. 211–230. doi:10.1007/978-3-0348-0859-0_13. ISBN 978-3-0348-0858-3. <a href="https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/abcLahoreProceedings.pdf" target="_blank">https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/abcLahoreProceedings.pdf</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Waldschmidt 2015. - Waldschmidt, Michel (2015). "Lecture on the abc Conjecture and Some of Its Consequences" (PDF). Mathematics in the 21st Century. Springer Proceedings in Mathematics & Statistics. Vol. 98. pp. 211–230. doi:10.1007/978-3-0348-0859-0_13. ISBN 978-3-0348-0858-3. <a href="https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/abcLahoreProceedings.pdf" target="_blank">https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/abcLahoreProceedings.pdf</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Elkies (1991). - Elkies, N. D. (1991). "ABC implies Mordell". International Mathematics Research Notices. 1991 (7): 99–109. doi:10.1155/S1073792891000144. <a href="https://doi.org/10.1155%2FS1073792891000144" target="_blank">https://doi.org/10.1155%2FS1073792891000144</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Van Frankenhuijsen (2002). - Van Frankenhuijsen, Machiel (2002). "The ABC conjecture implies Vojta's height inequality for curves". J. Number Theory. 95 (2): 289–302. doi:10.1006/jnth.2001.2769. MR 1924103. <a href="https://doi.org/10.1006%2Fjnth.2001.2769" target="_blank">https://doi.org/10.1006%2Fjnth.2001.2769</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Langevin (1993). - Langevin, M. (1993). "Cas d'égalité pour le théorème de Mason et applications de la conjecture abc". Comptes rendus de l'Académie des sciences (in French). 317 (5): 441–444. <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Silverman (1988). - Silverman, Joseph H. (1988). "Wieferich's criterion and the abc-conjecture". Journal of Number Theory. 30 (2): 226–237. doi:10.1016/0022-314X(88)90019-4. Zbl 0654.10019. <a href="https://doi.org/10.1016%2F0022-314X%2888%2990019-4" target="_blank">https://doi.org/10.1016%2F0022-314X%2888%2990019-4</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Nitaj (1996). - Nitaj, Abderrahmane (1996). "La conjecture abc". Enseign. Math. (in French). 42 (1–2): 3–24. <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Granville, Andrew; Tucker, Thomas (2002). "It's As Easy As abc" (PDF). Notices of the AMS. 49 (10): 1224–1231. <a href="https://www.ams.org/notices/200210/fea-granville.pdf" target="_blank">https://www.ams.org/notices/200210/fea-granville.pdf</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Pomerance (2008). - Pomerance, Carl (2008). "Computational Number Theory". The Princeton Companion to Mathematics. Princeton University Press. pp. 361–362. <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Granville & Stark (2000). - Granville, Andrew; Stark, H. (2000). "ABC implies no "Siegel zeros" for L-functions of characters with negative exponent" (PDF). Inventiones Mathematicae. 139 (3): 509–523. Bibcode:2000InMat.139..509G. doi:10.1007/s002229900036. S2CID 6901166. <a href="http://www.dms.umontreal.ca/~andrew/PDF/NoSiegelfinal.pdf" target="_blank">http://www.dms.umontreal.ca/~andrew/PDF/NoSiegelfinal.pdf</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">The ABC-conjecture, Frits Beukers, ABC-DAY, Leiden, Utrecht University, 9 September 2005. <a href="http://www.math.uu.nl/people/beukers/ABCpresentation.pdf" target="_blank">http://www.math.uu.nl/people/beukers/ABCpresentation.pdf</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">Mollin (2009); Mollin (2010, p. 297) - Mollin, R.A. (2009). "A note on the ABC-conjecture" (PDF). Far East Journal of Mathematical Sciences. 33 (3): 267–275. ISSN 0972-0871. Zbl 1241.11034. Archived from the original (PDF) on 2016-03-04. Retrieved 2013-06-14. <a href="https://web.archive.org/web/20160304053930/http://people.ucalgary.ca/~ramollin/abcconj.pdf" target="_blank">https://web.archive.org/web/20160304053930/http://people.ucalgary.ca/~ramollin/abcconj.pdf</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">Browkin (2000, p. 10) - Browkin, Jerzy (2000). "The abc-conjecture". In Bambah, R. P.; Dumir, V. C.; Hans-Gill, R. J. (eds.). Number Theory. Trends in Mathematics. Basel: Birkhäuser. pp. 75–106. ISBN 3-7643-6259-6. <a href="https://archive.org/details/numbertheory00bamb_636" target="_blank">https://archive.org/details/numbertheory00bamb_636</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Oesterlé 1988. - Oesterlé, Joseph (1988), "Nouvelles approches du "théorème" de Fermat", Astérisque, Séminaire Bourbaki exp 694 (161): 165–186, ISSN 0303-1179, MR 0992208 <a href="http://www.numdam.org/item?id=SB_1987-1988__30__165_0" target="_blank">http://www.numdam.org/item?id=SB_1987-1988__30__165_0</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">Granville (1998). - Granville, A. (1998). "ABC Allows Us to Count Squarefrees" (PDF). International Mathematics Research Notices. 1998 (19): 991–1009. doi:10.1155/S1073792898000592. <a href="http://www.dms.umontreal.ca/~andrew/PDF/polysq3.pdf" target="_blank">http://www.dms.umontreal.ca/~andrew/PDF/polysq3.pdf</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">Pasten, Hector (2017), "Definability of Frobenius orbits and a result on rational distance sets", Monatshefte für Mathematik, 182 (1): 99–126, doi:10.1007/s00605-016-0973-2, MR 3592123, S2CID 7805117 <a href="/wiki/Monatshefte_f%C3%BCr_Mathematik" target="_blank">/wiki/Monatshefte_f%C3%BCr_Mathematik</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
<li id="fn:28">arXiv:math/0408168 Andrea Surroca, Siegel’s theorem and the abc conjecture, Riv. Mat. Univ. Parma (7) 3, 2004, S. 323–332 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
<li id="fn:29">"Synthese resultaten", RekenMeeMetABC.nl (in Dutch), archived from the original on December 22, 2008, retrieved October 3, 2012. <a href="https://web.archive.org/web/20081222221716/http://rekenmeemetabc.nl/?item=h_stats" target="_blank">https://web.archive.org/web/20081222221716/http://rekenmeemetabc.nl/?item=h_stats</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></li>
<li id="fn:30">"Data collected sofar", ABC@Home, archived from the original on May 15, 2014, retrieved April 30, 2014 <a href="https://web.archive.org/web/20140515021303/http://abcathome.com/data/" target="_blank">https://web.archive.org/web/20140515021303/http://abcathome.com/data/</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></li>
<li id="fn:31">"100 unbeaten triples". Reken mee met ABC. 2010-11-07. <a href="http://www.math.leidenuniv.nl/~desmit/abc/index.php?set=2" target="_blank">http://www.math.leidenuniv.nl/~desmit/abc/index.php?set=2</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></li>
<li id="fn:32">Bombieri & Gubler (2006), p. 404. - Bombieri, Enrico; Gubler, Walter (2006). Heights in Diophantine Geometry. New Mathematical Monographs. Vol. 4. Cambridge University Press. ISBN 978-0-521-71229-3. Zbl 1130.11034. <a href="https://zbmath.org/?format=complete&q=an:1130.11034" target="_blank">https://zbmath.org/?format=complete&q=an:1130.11034</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></li>
<li id="fn:33">"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 Woit, Peter (May 26, 2007), "Proof of the abc Conjecture?", Not Even Wrong. <a href="/wiki/Peter_Woit" target="_blank">/wiki/Peter_Woit</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></li>
<li id="fn:34">Ball, Peter (10 September 2012). "Proof claimed for deep connection between primes". Nature. doi:10.1038/nature.2012.11378. Retrieved 19 March 2018. <a href="https://www.nature.com/news/proof-claimed-for-deep-connection-between-primes-1.11378" target="_blank">https://www.nature.com/news/proof-claimed-for-deep-connection-between-primes-1.11378</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></li>
<li id="fn:35">Mochizuki, Shinichi (4 March 2021). "Inter-universal Teichmüller Theory IV: Log-Volume Computations and Set-Theoretic Foundations". Publications of the Research Institute for Mathematical Sciences. 57 (1): 627–723. doi:10.4171/PRIMS/57-1-4. S2CID 3135393. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:35" class="footnote-back-ref">↩</a></li>
<li id="fn:36">Calegari, Frank (December 17, 2017). "The ABC conjecture has (still) not been proved". Retrieved March 17, 2018. <a href="/wiki/Frank_Calegari" target="_blank">/wiki/Frank_Calegari</a> <a href="#fnref:36" class="footnote-back-ref">↩</a></li>
<li id="fn:37">Revell, Timothy (September 7, 2017). "Baffling ABC maths proof now has impenetrable 300-page 'summary'". New Scientist. <a href="https://www.newscientist.com/article/2146647-baffling-abc-maths-proof-now-has-impenetrable-300-page-summary/" target="_blank">https://www.newscientist.com/article/2146647-baffling-abc-maths-proof-now-has-impenetrable-300-page-summary/</a> <a href="#fnref:37" class="footnote-back-ref">↩</a></li>
<li id="fn:38">Scholze, Peter; Stix, Jakob. "Why abc is still a conjecture" (PDF). Archived from the original (PDF) on February 8, 2020. Retrieved September 23, 2018. (updated version of their May report Archived 2020-02-08 at the Wayback Machine) <a href="/wiki/Peter_Scholze" target="_blank">/wiki/Peter_Scholze</a> <a href="#fnref:38" class="footnote-back-ref">↩</a></li>
<li id="fn:39">Fesenko, Ivan (28 September 2016). "Fukugen". Inference. 2 (3). Retrieved 30 October 2021. <a href="/wiki/Ivan_Fesenko" target="_blank">/wiki/Ivan_Fesenko</a> <a href="#fnref:39" class="footnote-back-ref">↩</a></li>
<li id="fn:40">Conrad, Brian (December 15, 2015). "Notes on the Oxford IUT workshop by Brian Conrad". Retrieved March 18, 2018. <a href="/wiki/Brian_Conrad" target="_blank">/wiki/Brian_Conrad</a> <a href="#fnref:40" class="footnote-back-ref">↩</a></li>
<li id="fn:41">Castelvecchi, Davide (8 October 2015). "The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof". Nature. 526 (7572): 178–181. Bibcode:2015Natur.526..178C. doi:10.1038/526178a. PMID 26450038. <a href="https://doi.org/10.1038%2F526178a" target="_blank">https://doi.org/10.1038%2F526178a</a> <a href="#fnref:41" class="footnote-back-ref">↩</a></li>
<li id="fn:42">Klarreich, Erica (September 20, 2018). "Titans of Mathematics Clash Over Epic Proof of ABC Conjecture". Quanta Magazine. <a href="/wiki/Erica_Klarreich" target="_blank">/wiki/Erica_Klarreich</a> <a href="#fnref:42" class="footnote-back-ref">↩</a></li>
<li id="fn:43">"March 2018 Discussions on IUTeich". Retrieved October 2, 2018. Web-page by Mochizuki describing discussions and linking consequent publications and supplementary material <a href="http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTch-discussions-2018-03.html" target="_blank">http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTch-discussions-2018-03.html</a> <a href="#fnref:43" class="footnote-back-ref">↩</a></li>
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<li id="fn:45">Mochizuki, Shinichi. "Report on Discussions, Held during the Period March 15 – 20, 2018, Concerning Inter-Universal Teichmüller Theory" (PDF). Retrieved February 1, 2019. the ... discussions ... constitute the first detailed, ... substantive discussions concerning negative positions ... IUTch. <a href="/wiki/Shinichi_Mochizuki" target="_blank">/wiki/Shinichi_Mochizuki</a> <a href="#fnref:45" class="footnote-back-ref">↩</a></li>
<li id="fn:46">Mochizuki, Shinichi (July 2018). "Comments on the manuscript by Scholze-Stix concerning Inter-Universal Teichmüller Theory" (PDF). S2CID 174791744. Retrieved October 2, 2018. <a href="/wiki/Shinichi_Mochizuki" target="_blank">/wiki/Shinichi_Mochizuki</a> <a href="#fnref:46" class="footnote-back-ref">↩</a></li>
<li id="fn:47">Mochizuki, Shinichi. "Comments on the manuscript (2018-08 version) by Scholze-Stix concerning Inter-Universal Teichmüller Theory" (PDF). Retrieved October 2, 2018. <a href="/wiki/Shinichi_Mochizuki" target="_blank">/wiki/Shinichi_Mochizuki</a> <a href="#fnref:47" class="footnote-back-ref">↩</a></li>
<li id="fn:48">Castelvecchi, Davide (9 April 2020). "Mathematical proof that rocked number theory will be published". Nature. 580 (7802): 177. Bibcode:2020Natur.580..177C. doi:10.1038/d41586-020-00998-2. PMID 32246118. S2CID 214786566. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:48" class="footnote-back-ref">↩</a></li>
<li id="fn:49">Castelvecchi, Davide (9 April 2020). "Mathematical proof that rocked number theory will be published". Nature. 580 (7802): 177. Bibcode:2020Natur.580..177C. doi:10.1038/d41586-020-00998-2. PMID 32246118. S2CID 214786566. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:49" class="footnote-back-ref">↩</a></li>
<li id="fn:50">Mochizuki, Shinichi. "Mochizuki's proof of ABC conjecture". Retrieved July 13, 2021. <a href="/wiki/Shinichi_Mochizuki" target="_blank">/wiki/Shinichi_Mochizuki</a> <a href="#fnref:50" class="footnote-back-ref">↩</a></li>
</ol>

abc conjecture open-in-new

abc conjecture