The abc conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory that arose out of a discussion of Joseph Oesterlé and David Masser in 1985.^{[1]}^{[2]} It is stated in terms of three positive integersa, b and c (hence the name) that are relatively prime and satisfy a + b = c. The conjecture essentially states that the product of the distinct prime factors of abc is usually not much smaller than c. A number of famous conjectures and theorems in number theory would follow immediately from the abc conjecture or its versions. Mathematician Dorian Goldfeld described the abc conjecture as "The most important unsolved problem in Diophantine analysis".^{[3]}
The abc conjecture originated as the outcome of attempts by Oesterlé and Masser to understand the Szpiro conjecture about elliptic curves,^{[4]} 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.^{[1]}
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 regarded as unproven.^{[5]}^{[6]}
FormulationsEdit
Before stating the conjecture, the notion of the radical of an integer must be introduced: for a positive integern, 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:
For every positive real numberε, there exist only finitely many triples (a, b, c) of coprime positive integers, with a + b = c, such that^{[7]}
$c>\operatorname {rad} (abc)^{1+\varepsilon }.$
An equivalent formulation is:
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:^{[7]}
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. The third 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).
Examples of triples with small radicalEdit
The condition that ε > 0 is necessary as there exist infinitely many triples a, b, c with c > rad(abc). For example, let
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
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, the following results:
$\operatorname {rad} (abc)<{\tfrac {2}{p}}c.$
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 = 3^{10}·109 = 6436341,
c = 23^{5} = 6436343,
rad(abc) = 15042.
Some consequencesEdit
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 conditional proof. The consequences include:
The L-functionL(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.^{[15]}
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 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}.^{[17]}
As equivalent, the modified Szpiro conjecture, which would yield a bound of rad(abc)^{1.2+ε}.^{[1]}
Fermat's Last Theorem has a famously difficult proof by Andrew Wiles. However it follows easily, at least for $n\geq 6$, 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 $n\geq 6$.^{[19]}
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 non-torsion rational point of an elliptic curve.
A negative solution to the Erdős–Ulam problem on dense sets of Euclidean points with rational distances.^{[20]}
Theoretical resultsEdit
The abc conjecture implies that c can be bounded above by a near-linear function of the radical of abc. Bounds are known that are exponential. Specifically, the following bounds have been proven:
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 resultsEdit
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.
Distribution of triples with q > 1^{[21]}
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.^{[22]}
Highest-quality triples^{[23]}
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 qualityq(a, b, c) of the triple (a, b, c) is defined above.
Refined forms, generalizations and related statementsEdit
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.^{[24]}
Andrew Granville noticed that the minimum of the function ${\big (}\varepsilon ^{-\omega }\operatorname {rad} (abc){\big )}^{1+\varepsilon }$ over $\varepsilon >0$ occurs when $\varepsilon ={\frac {\omega }{\log {\big (}\operatorname {rad} (abc){\big )}}}.$
This inspired Baker (2004) to propose a sharper form of the abc conjecture, namely:
with κ an absolute constant. After some computational experiments he found that a value of $6/5$ was admissible for κ. This version is called the "explicit abc conjecture".
Baker (1998) also describes related conjectures of Andrew Granville that would give upper bounds on c of the form
$K^{\Omega (abc)}\operatorname {rad} (abc),$
where Ω(n) is the total number of prime factors of n, and
Lucien Szpiro proposed a solution in 2007, but it was found to be incorrect shortly afterwards.^{[25]}
Since August 2012, Shinichi Mochizuki has claimed a proof of Szpiro's conjecture and therefore the abc conjecture.^{[26]} He released a series of four preprints developing a new theory he called inter-universal Teichmüller theory (IUTT), which is then applied to prove the abc conjecture.^{[27]}
The papers have not been accepted by the mathematical community as providing a proof of abc.^{[28]} This is not only because of their length and the difficulty of understanding them,^{[29]} but also because at least one specific point in the argument has been identified as a gap by some other experts.^{[30]} Although a few mathematicians have vouched for the correctness of the proof,^{[31]} and have attempted to communicate their understanding via workshops on IUTT, they have failed to convince the number theory community at large.^{[32]}^{[33]}
In March 2018, Peter Scholze and Jakob Stix visited Kyoto for discussions with Mochizuki.^{[34]}^{[35]}
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";^{[30]}
Mochizuki claimed that they misunderstood vital aspects of the theory and made invalid simplifications.^{[36]}^{[37]}^{[38]}
On April 3, 2020, two mathematicians from the Kyoto research institute 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.^{[5]} 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".^{[5]} In March 2021, Mochizuki's proof was published in RIMS.^{[39]}
^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}^{b}^{c}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. PMID32246118.
^Further comment by P. Scholze at Not Even Wrong math.columbia.edu^{[self-published source?]}
^Granville, Andrew; Tucker, Thomas (2002). "It's As Easy As abc" (PDF). Notices of the AMS. 49 (10): 1224–1231.
^
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
^"Synthese resultaten", RekenMeeMetABC.nl (in Dutch), archived from the original on December 22, 2008, retrieved October 3, 2012.
^"Data collected sofar", ABC@Home, archived from the original on May 15, 2014, retrieved April 30, 2014
^"100 unbeaten triples". Reken mee met ABC. 2010-11-07.
^"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.
^
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 (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.
^Calegari, Frank (December 17, 2017). "The ABC conjecture has (still) not been proved". Retrieved March 17, 2018.
^Revell, Timothy (September 7, 2017). "Baffling ABC maths proof now has impenetrable 300-page 'summary'". New Scientist.
^ ^{a}^{b}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)
^Fesenko, Ivan (28 September 2016). "Fukugen". Inference. 2 (3). Retrieved 30 October 2021.
^Conrad, Brian (December 15, 2015). "Notes on the Oxford IUT workshop by Brian Conrad". Retrieved March 18, 2018.
^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. PMID26450038.
^
"March 2018 Discussions on IUTeich". Retrieved October 2, 2018. Web-page by Mochizuki describing discussions and linking consequent publications and supplementary material
^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.
^Mochizuki, Shinichi (July 2018). "Comments on the manuscript by Scholze-Stix concerning Inter-Universal Teichmüller Theory" (PDF). S2CID 174791744. Retrieved October 2, 2018.
^Mochizuki, Shinichi. "Comments on the manuscript (2018-08 version) by Scholze-Stix concerning Inter-Universal Teichmüller Theory" (PDF). Retrieved October 2, 2018.
^Mochizuki, Shinichi. "Mochizuki's proof of ABC conjecture". Retrieved July 13, 2021.
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