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abc conjecture
Field  Number theory 

Conjectured by  Joseph Oesterlé David Masser 
Conjectured in  1985 
Equivalent to  Modified Szpiro conjecture 
Consequences 
The abc conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory, first proposed by Joseph Oesterlé (1988) and David Masser (1985). It is stated in terms of three positive integers, a, b and c (hence the name) that are relatively prime and satisfy a + b = c. If d denotes the product of the distinct prime factors 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. A number of famous conjectures and theorems in number theory would follow immediately from the abc conjecture or its versions. Goldfeld (1996) described the abc conjecture as "the most important unsolved problem in Diophantine analysis".
The abc conjecture originated as the outcome of attempts by Oesterlé and Masser to understand the Szpiro conjecture about elliptic curves,^{[1]} which involves more geometric structures in its statement than the abc conjecture. The abc conjecture was shown to be equivalent to the modified Szpiro's conjecture.^{[2]}
Various attempts to prove the abc conjecture have been made, but none are currently accepted by the mainstream mathematical community and as of 2020, the conjecture is still largely regarded as unproven.^{[3]}^{[4]}
Formulations[edit source  edit]
Before we state the conjecture we introduce the notion of the radical of an integer: for a positive integer n, the radical of n, denoted rad(n), is the product of the distinct prime factors of n. For example
 rad(16) = rad(2^{4}) = rad(2) = 2,
 rad(17) = 17,
 rad(18) = rad(2 ⋅ 3^{2}) = 2 · 3 = 6,
 rad(1000000) = rad(2^{6} ⋅ 5^{6}) = 2 ⋅ 5 = 10.
If a, b, and c are coprime^{[notes 1]} 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:
 ABC conjecture. 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
 <math>c > \operatorname{rad}(abc)^{1+\varepsilon}.</math>
An equivalent formulation:
 ABC conjecture II. For every positive real number ε, there exists a constant K_{ε} such that for all triples (a, b, c) of coprime positive integers, with a + b = c:
 <math>c < K_{\varepsilon} \cdot \operatorname{rad}(abc)^{1+\varepsilon}.</math>
A third equivalent formulation of the conjecture involves the quality q(a, b, c) of the triple (a, b, c), defined as
 <math> q(a, b, c) = \frac{ \log(c) }{ \log\big(\operatorname{rad}(abc)\big) }.</math>
For example:
 q(4, 127, 131) = log(131) / log(rad(4·127·131)) = log(131) / log(2·127·131) = 0.46820...
 q(3, 125, 128) = log(128) / log(rad(3·125·128)) = log(128) / log(30) = 1.426565...
A typical triple (a, b, c) of coprime positive integers with a + b = c will have c < rad(abc), i.e. q(a, b, c) < 1. Triples with q > 1 such as in the second example are rather special, they consist of numbers divisible by high powers of small prime numbers.
 ABC conjecture III. 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) .
Examples of triples with small radical[edit source  edit]
The condition that ε > 0 is necessary as there exist infinitely many triples a, b, c with c > rad(abc). For example, let
 <math>a = 1, \quad b = 2^{6n}  1, \quad c = 2^{6n}, \qquad n > 1.</math>
The integer b is divisible by 9:
 <math> b = 2^{6n}  1 = 64^n  1 = (64  1) (\cdots) = 9 \cdot 7 \cdot (\cdots).</math>
Using this fact we calculate:
 <math>\begin{align}
\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} \\ &= 2 \tfrac{b}{3} \\ &< \tfrac{2}{3} c. \end{align}</math>
By replacing the exponent 6n by 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
 <math>a = 1, \quad b = 2^{p(p1)n}  1, \quad c = 2^{p(p1)n}, \qquad n > 1.</math>
Now we claim that b is divisible by p^{2}:
 <math>\begin{align}
b &= 2^{p(p1)n}  1 \\ &= \left(2^{p(p1)}\right)^n  1 \\ &= \left(2^{p(p1)}  1\right) (\cdots) \\ &= p^2 \cdot r (\cdots). \end{align}</math>
The last step uses the fact that p^{2} divides 2^{p(p−1)} − 1. This follows from Fermat's little theorem, which shows that, for p > 2, 2^{p−1} = pk + 1 for some integer k. Raising both sides to the power of p then shows that 2^{p(p−1)} = p^{2}(...) + 1.
And now with a similar calculation as above we have
 <math>\operatorname{rad}(abc) < \tfrac{2}{p} c.</math>
A list of the highestquality 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 = 3^{10}·109 = 6436341,
 c = 23^{5} = 6436343,
 rad(abc) = 15042.
Some consequences[edit source  edit]
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. The consequences include:
 Roth's theorem on diophantine approximation of algebraic numbers.^{[5]}
 The Mordell conjecture (already proven in general by Gerd Faltings).^{[6]}
 As equivalent, Vojta's conjecture in dimension 1.^{[7]}
 The Erdős–Woods conjecture allowing for a finite number of counterexamples.^{[8]}
 The existence of infinitely many nonWieferich primes in every base b > 1.^{[9]}
 The weak form of Marshall Hall's conjecture on the separation between squares and cubes of integers.^{[10]}
 The Fermat–Catalan conjecture, a generalization of Fermat's last theorem concerning powers that are sums of powers.^{[11]}
 The Lfunction L(s, χ_{d}) formed with the Legendre symbol, has no Siegel zero, given a uniform version of the abc conjecture in number fields, not just the abc conjecture as formulated above for rational integers.^{[12]}
 A polynomial P(x) has only finitely many perfect powers for all integers x if P has at least three simple zeros.^{[13]}
 A generalization of Tijdeman's theorem concerning the number of solutions of y^{m} = x^{n} + k (Tijdeman's theorem answers the case k = 1), and Pillai's conjecture (1931) concerning the number of solutions of Ay^{m} = Bx^{n} + k.
 As equivalent, the Granville–Langevin conjecture, that if f is a squarefree 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}.^{[14]}
 As equivalent, the modified Szpiro conjecture, which would yield a bound of rad(abc)^{1.2+ε}.^{[2]}
 Dąbrowski (1996) has shown that the abc conjecture implies that the Diophantine equation n! + A = k^{2} has only finitely many solutions for any given integer A.
 There are ~c_{f}N positive integers n ≤ N for which f(n)/B' is squarefree, with c_{f} > 0 a positive constant defined as:^{[15]}
 <math>c_f = \prod_{\text{prime }p} x_i \left ( 1  \frac{\omega\,\!_f (p)}{p^{2+q_p}} \right ).</math>
 Fermat's Last Theorem has a famously difficult proof by Andrew Wiles. However it follows easily, at least for <math>n \ge 6</math>, from an effective form of a weak version of the abc conjecture. The abc conjecture says the lim sup 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 <math>n \ge 6</math>.^{[16]}
 The Beal conjecture, a generalization of Fermat's last theorem proposing that if A, B, C, x, y, and z are positive integers with A^{x} + B^{y} = C^{z} 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.
 Lang’s conjecture, a lower bound for the height of a nontorsion rational point of an elliptic curve.
Theoretical results[edit source  edit]
The abc conjecture implies that c can be bounded above by a nearlinear function of the radical of abc. Bounds are known that are exponential. Specifically, the following bounds have been proven:
 <math>c < \exp{ \left(K_1 \operatorname{rad}(abc)^{15}\right) } </math> (Stewart & Tijdeman 1986),
 <math>c < \exp{ \left(K_2 \operatorname{rad}(abc)^{\frac{2}{3} + \varepsilon}\right) } </math> (Stewart & Yu 1991), and
 <math>c < \exp{ \left(K_3 \operatorname{rad}(abc)^{\frac{1}{3}}\left(\log(\operatorname{rad}(abc))\right)^3\right) } </math> (Stewart & Yu 2001).
In these bounds, K_{1} and K_{3} are constants that do not depend on a, b, or c, and K_{2} is a constant that depends on ε (in an effectively computable way) but not on a, b, or c. The bounds apply to any triple for which c > 2.
Computational results[edit source  edit]
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.
q c

q > 1  q > 1.05  q > 1.1  q > 1.2  q > 1.3  q > 1.4 

c < 10^{2}  6  4  4  2  0  0 
c < 10^{3}  31  17  14  8  3  1 
c < 10^{4}  120  74  50  22  8  3 
c < 10^{5}  418  240  152  51  13  6 
c < 10^{6}  1,268  667  379  102  29  11 
c < 10^{7}  3,499  1,669  856  210  60  17 
c < 10^{8}  8,987  3,869  1,801  384  98  25 
c < 10^{9}  22,316  8,742  3,693  706  144  34 
c < 10^{10}  51,677  18,233  7,035  1,159  218  51 
c < 10^{11}  116,978  37,612  13,266  1,947  327  64 
c < 10^{12}  252,856  73,714  23,773  3,028  455  74 
c < 10^{13}  528,275  139,762  41,438  4,519  599  84 
c < 10^{14}  1,075,319  258,168  70,047  6,665  769  98 
c < 10^{15}  2,131,671  463,446  115,041  9,497  998  112 
c < 10^{16}  4,119,410  812,499  184,727  13,118  1,232  126 
c < 10^{17}  7,801,334  1,396,909  290,965  17,890  1,530  143 
c < 10^{18}  14,482,065  2,352,105  449,194  24,013  1,843  160 
As of May 2014, ABC@Home had found 23.8 million triples.^{[18]}
Rank  q  a  b  c  Discovered by 

1  1.6299  2  3^{10}·109  23^{5}  Eric Reyssat 
2  1.6260  11^{2}  3^{2}·5^{6}·7^{3}  2^{21}·23  Benne de Weger 
3  1.6235  19·1307  7·29^{2}·31^{8}  2^{8}·3^{22}·5^{4}  Jerzy Browkin, Juliusz Brzezinski 
4  1.5808  283  5^{11}·13^{2}  2^{8}·3^{8}·17^{3}  Jerzy Browkin, Juliusz Brzezinski, Abderrahmane Nitaj 
5  1.5679  1  2·3^{7}  5^{4}·7  Benne de Weger 
Note: the quality q(a, b, c) of the triple (a, b, c) is defined above.
[edit source  edit]
The abc conjecture is an integer analogue of the Mason–Stothers theorem 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.^{[20]}
Andrew Granville noticed that the minimum of the function <math>\big(\varepsilon^{\omega}\operatorname{rad}(abc)\big)^{1+\varepsilon}</math> over <math>\varepsilon > 0</math> occurs when <math>\varepsilon = \frac{\omega}{\log\big(\operatorname{rad}(abc)\big)}.</math>
This incited Baker (2004) to propose a sharper form of the abc conjecture, namely:
 <math>c < \kappa \operatorname{rad}(abc) \frac{\Big(\log\big(\operatorname{rad}(abc)\big)\Big)^\omega}{\omega!}</math>
with κ an absolute constant. After some computational experiments he found that a value of <math>6/5</math> was admissible for κ.
This version is called "explicit abc conjecture".
Baker (1998) also describes related conjectures of Andrew Granville that would give upper bounds on c of the form
 <math>K^{\Omega(a b c)} \operatorname{rad}(a b c),</math>
where Ω(n) is the total number of prime factors of n, and
 <math>O\big(\operatorname{rad}(a b c) \Theta(a b c)\big),</math>
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 C_{1} such that
 <math>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)</math>
holds whereas there is a constant C_{2} such that
 <math>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)</math>
holds infinitely often.
Browkin & Brzeziński (1994) formulated the n conjecture—a version of the abc conjecture involving n > 2 integers.
Claimed proofs[edit source  edit]
Lucien Szpiro proposed a solution in 2007, but it was found to be incorrect shortly afterwards.^{[21]}
In August 2012, Shinichi Mochizuki claimed a proof of Szpiro's conjecture and therefore the abc conjecture.^{[22]} He released a series of four preprints developing a new theory called interuniversal Teichmüller theory (IUTT) which is then applied to prove several famous conjectures in number theory, including the abc conjecture and the hyperbolic Vojta's conjecture.^{[23]} The papers have not been accepted by the mathematical community as providing a proof of abc.^{[24]} This is not only because of their difficulty to understand and length,^{[25]} but also because at least one specific point in the argument has been identified as a gap by some other experts.^{[26]} Although a few mathematicians have vouched for the correctness of the proof,^{[27]} and have attempted to communicate their understanding via workshops on IUTT, they have failed to convince the number theory community at large.^{[28]}^{[29]}
In March 2018, Peter Scholze and Jakob Stix visited Kyoto for discussions with Mochizuki.^{[30]}^{[31]} While they did not resolve the differences, they brought them into clearer focus. Scholze and Stix concluded that the gap was "so severe that … small modifications will not rescue the proof strategy";^{[32]} Mochizuki claimed that they misunderstood vital aspects of the theory and made invalid simplifications.^{[33]}^{[34]}^{[35]}
On April 3, 2020, two Japanese mathematicians announced that Mochizuki's claimed proof would be published in Publications of the Research Institute for Mathematical Sciences (RIMS), a journal of which Mochizuki is chief editor.^{[3]} The announcement was received with skepticism by Kiran Kedlaya and Edward Frenkel, as well as being described by Nature as "unlikely to move many researchers over to Mochizuki's camp."^{[3]}
See also[edit source  edit]
Notes[edit source  edit]
 ↑ When a + b = c, coprimeness of a, b, c implies pairwise coprimeness of a, b, c. So in this case, it does not matter which concept we use.
References[edit source  edit]
 ↑ Fesenko, Ivan (2015), "Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki" (PDF), European Journal of Mathematics, 1 (3): 405–440, doi:10.1007/s4087901500660.
 ↑ ^{2.0} ^{2.1} Oesterlé (1988).
 ↑ ^{3.0} ^{3.1} ^{3.2} Castelvecchi, Davide (3 April 2020). "Mathematical proof that rocked number theory will be published". Nature. doi:10.1038/d41586020009982.
 ↑ Further comment by P. Scholze at Not Even Wrong.
 ↑ Bombieri (1994).
 ↑ Elkies (1991).
 ↑ Van Frankenhuijsen (2002).
 ↑ Langevin (1993).
 ↑ Silverman (1988).
 ↑ Nitaj (1996).
 ↑ Pomerance (2008).
 ↑ Granville & Stark (2000).
 ↑ The ABCconjecture, Frits Beukers, ABCDAY, Leiden, Utrecht University, 9 September 2005.
 ↑ Mollin (2009); Mollin (2010, p. 297)
 ↑ Granville (1998).
 ↑ Granville, Andrew; Tucker, Thomas (2002). "It's As Easy As abc" (PDF). Notices of the AMS. 49 (10): 1224–1231.
 ↑ "Synthese resultaten", RekenMeeMetABC.nl (in Dutch), archived from the original on December 22, 2008, retrieved October 3, 2012 CS1 maint: discouraged parameter (link).
 ↑ "Data collected sofar", ABC@Home, archived from the original on May 15, 2014, retrieved April 30, 2014 CS1 maint: discouraged parameter (link)
 ↑ "100 unbeaten triples". Reken mee met ABC. 20101107.
 ↑ Bombieri & Gubler (2006), p. 404.
 ↑ "Finiteness Theorems for Dynamical Systems", Lucien Szpiro, talk at Conference on Lfunctions 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 CS1 maint: discouraged parameter (link).
 ↑ Ball, Peter (10 September 2012). "Proof claimed for deep connection between primes". Nature. doi:10.1038/nature.2012.11378. Retrieved 19 March 2018.
 ↑ Mochizuki, Shinichi (May 2015). Interuniversal Teichmuller Theory IV: Logvolume Computations and Settheoretic Foundations, available at http://www.kurims.kyotou.ac.jp/~motizuki/papersenglish.html
 ↑ "The ABC conjecture has still not been proved". December 17, 2017. Retrieved March 17, 2018.
 ↑ Revell, Timothy (September 7, 2017). "Baffling ABC maths proof now has impenetrable 300page 'summary'". New Scientist.
 ↑ "The ABC conjecture has still not been proved, comment by Bcnrd". December 22, 2017. Retrieved March 18, 2017.
 ↑ Fesenko, Ivan. "Fukugen". Inference. Retrieved 19 March 2018. CS1 maint: discouraged parameter (link)
 ↑ Conrad, Brian (December 15, 2015). "Notes on the Oxford IUT workshop by Brian Conrad". Retrieved March 18, 2018. CS1 maint: discouraged parameter (link)
 ↑ 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.
 ↑ Klarreich, Erica (September 20, 2018). "Titans of Mathematics Clash Over Epic Proof of ABC Conjecture". Quanta Magazine. CS1 maint: discouraged parameter (link)
 ↑ "March 2018 Discussions on IUTeich". Retrieved October 2, 2018. Webpage by Mochizuki describing discussions and linking consequent publications and supplementary material
 ↑ Scholze, Peter; Stix, Jakob. "Why abc is still a conjecture" (PDF). Retrieved September 23, 2018. CS1 maint: discouraged parameter (link) (updated version of their May report)
 ↑
Mochizuki, Shinichi. "Report on Discussions, Held during the Period March 15 – 20, 2018, Concerning InterUniversal Teichmüller Theory" (PDF). Retrieved February 1, 2019.
the … discussions … constitute the first detailed, … substantive discussions concerning negative positions … IUTch.
CS1 maint: discouraged parameter (link)  ↑ Mochizuki, Shinichi. "Comments on the manuscript by ScholzeStix concerning InterUniversal Teichmüller Theory" (PDF). Retrieved October 2, 2018. CS1 maint: discouraged parameter (link)
 ↑ Mochizuki, Shinichi. "Comments on the manuscript (201808 version) by ScholzeStix concerning InterUniversal Teichmüller Theory" (PDF). Retrieved October 2, 2018. CS1 maint: discouraged parameter (link)
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 Pomerance, Carl (2008). "Computational Number Theory". The Princeton Companion to Mathematics. Princeton University Press. pp. 361–362. CS1 maint: discouraged parameter (link) CS1 maint: ref=harv (link)
 Silverman, Joseph H. (1988). "Wieferich's criterion and the abcconjecture". Journal of Number Theory. 30 (2): 226–237. doi:10.1016/0022314X(88)900194. Zbl 0654.10019. CS1 maint: discouraged parameter (link) CS1 maint: ref=harv (link)
 Robert, Olivier; Stewart, Cameron L.; Tenenbaum, Gérald (2014). "A refinement of the abc conjecture" (PDF). Bulletin of the London Mathematical Society. 46 (6): 1156–1166. doi:10.1112/blms/bdu069.CS1 maint: ref=harv (link)
 Robert, Olivier; Tenenbaum, Gérald (2013). "Sur la répartition du noyau d'un entier". Indag. Math. 24 (4): 802–914. doi:10.1016/j.indag.2013.07.007.CS1 maint: ref=harv (link)
 Stewart, C. L.; Tijdeman, R. (1986). "On the OesterléMasser conjecture". Monatshefte für Mathematik. 102 (3): 251–257. doi:10.1007/BF01294603. CS1 maint: discouraged parameter (link) CS1 maint: ref=harv (link)
 Stewart, C. L.; Yu, Kunrui (1991). "On the abc conjecture". Mathematische Annalen. 291 (1): 225–230. doi:10.1007/BF01445201. CS1 maint: discouraged parameter (link) CS1 maint: ref=harv (link)
 Stewart, C. L.; Yu, Kunrui (2001). "On the abc conjecture, II". Duke Mathematical Journal. 108 (1): 169–181. doi:10.1215/S0012709401108156. CS1 maint: discouraged parameter (link) CS1 maint: ref=harv (link)
 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.CS1 maint: ref=harv (link)
External links[edit source  edit]
 ABC@home Distributed computing project called ABC@Home.
 Easy as ABC: Easy to follow, detailed explanation by Brian Hayes.
 Weisstein, Eric W. "abc Conjecture". MathWorld.
 Abderrahmane Nitaj's ABC conjecture home page
 Bart de Smit's ABC Triples webpage
 http://www.math.columbia.edu/~goldfeld/ABCConjecture.pdf
 The ABC's of Number Theory by Noam D. Elkies
 Questions about Number by Barry Mazur
 Philosophy behind Mochizuki’s work on the ABC conjecture on MathOverflow
 ABC Conjecture Polymath project wiki page linking to various sources of commentary on Mochizuki's papers.
 abc Conjecture Numberphile video
 News about IUT by Mochizuki
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