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Abc conjecture

## Summary

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 integers a, 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]

Field Number theory 1985 Modified Szpiro conjecture
Mathematician Joseph Oesterlé
Mathematician David Masser

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]

## Formulations

Before stating the conjecture, the notion of the radical of an integer must be introduced: for a positive integer n, the radical of n, denoted rad(n), is the product of the distinct prime factors of n. For example

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 number ε, there exist only finitely many triples (a, b, c) of coprime positive integers, with a + b = c, such that[7]
${\displaystyle c>\operatorname {rad} (abc)^{1+\varepsilon }.}$

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:[7]
${\displaystyle c

Equivalently (using little o from big O notation):

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

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

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 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 radical

The condition that ε > 0 is necessary as there exist infinitely many triples a, b, c with c > rad(abc). For example, let

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

The integer b is divisible by 9:

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

Using this fact, the following calculation is made:

{\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}}\\&=2{\tfrac {b}{3}}\\&<{\tfrac {2}{3}}c.\end{aligned}}}

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

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

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

{\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 Fermat's little theorem, 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:

${\displaystyle \operatorname {rad} (abc)<{\tfrac {2}{p}}c.}$

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,

## Some consequences

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:

• Roth's theorem on Diophantine approximation of algebraic numbers.[8][7]
• The Mordell conjecture (already proven in general by Gerd Faltings).[9]
• As equivalent, Vojta's conjecture in dimension 1.[10]
• The Erdős–Woods conjecture allowing for a finite number of counterexamples.[11]
• The existence of infinitely many non-Wieferich primes in every base b > 1.[12]
• The weak form of Marshall Hall's conjecture on the separation between squares and cubes of integers.[13]
• The Fermat–Catalan conjecture, a generalization of Fermat's Last Theorem concerning powers that are sums of powers.[14]
• The L-function L(s, χd) 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 polynomial P(x) has only finitely many perfect powers for all integers x if P has at least three simple zeros.[16]
• A generalization of Tijdeman's theorem 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.
• 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|}nβ.[17]
• As equivalent, the modified Szpiro conjecture, which would yield a bound of rad(abc)1.2+ε.[1]
• Dąbrowski (1996) has shown that the abc conjecture implies that the Diophantine equation n! + A = k2 has only finitely many solutions for any given integer A.
• There are ~cfN positive integers nN for which f(n)/B' is square-free, with cf > 0 a positive constant defined as:[18]
${\displaystyle c_{f}=\prod _{{\text{prime }}p}x_{i}\left(1-{\frac {\omega \,\!_{f}(p)}{p^{2+q_{p}}}}\right).}$
• Fermat's Last Theorem has a famously difficult proof by Andrew Wiles. However it follows easily, at least for ${\displaystyle n\geq 6}$ , 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 ${\displaystyle n\geq 6}$ .[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 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.
• 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 results

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:

${\displaystyle c<\exp {\left(K_{1}\operatorname {rad} (abc)^{15}\right)}}$  (Stewart & Tijdeman 1986),
${\displaystyle c<\exp {\left(K_{2}\operatorname {rad} (abc)^{{\frac {2}{3}}+\varepsilon }\right)}}$  (Stewart & Yu 1991), and
${\displaystyle c<\exp {\left(K_{3}\operatorname {rad} (abc)^{\frac {1}{3}}\left(\log(\operatorname {rad} (abc)\right)^{3}\right)}}$  (Stewart & Yu 2001).

In these bounds, K1 and K3 are constants that do not depend on a, b, or c, and K2 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

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 < 102 6 4 4 2 0 0
c < 103 31 17 14 8 3 1
c < 104 120 74 50 22 8 3
c < 105 418 240 152 51 13 6
c < 106 1,268 667 379 102 29 11
c < 107 3,499 1,669 856 210 60 17
c < 108 8,987 3,869 1,801 384 98 25
c < 109 22,316 8,742 3,693 706 144 34
c < 1010 51,677 18,233 7,035 1,159 218 51
c < 1011 116,978 37,612 13,266 1,947 327 64
c < 1012 252,856 73,714 23,773 3,028 455 74
c < 1013 528,275 139,762 41,438 4,519 599 84
c < 1014 1,075,319 258,168 70,047 6,665 769 98
c < 1015 2,131,671 463,446 115,041 9,497 998 112
c < 1016 4,119,410 812,499 184,727 13,118 1,232 126
c < 1017 7,801,334 1,396,909 290,965 17,890 1,530 143
c < 1018 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 2 310·109 235 Eric Reyssat
2 1.6260 112 32·56·73 221·23 Benne de Weger
3 1.6235 19·1307 7·292·318 28·322·54 Jerzy Browkin, Juliusz Brzezinski
4 1.5808 283 511·132 28·38·173 Jerzy Browkin, Juliusz Brzezinski, Abderrahmane Nitaj
5 1.5679 1 2·37 54·7 Benne de Weger

Note: the quality q(a, b, c) of the triple (a, b, c) is defined above.

## Refined forms, generalizations and related statements

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

where ω is the total number of distinct primes dividing a, b and c.[24]

Andrew Granville noticed that the minimum of the function ${\displaystyle {\big (}\varepsilon ^{-\omega }\operatorname {rad} (abc){\big )}^{1+\varepsilon }}$  over ${\displaystyle \varepsilon >0}$  occurs when ${\displaystyle \varepsilon ={\frac {\omega }{\log {\big (}\operatorname {rad} (abc){\big )}}}.}$

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

${\displaystyle c<\kappa \operatorname {rad} (abc){\frac {{\Big (}\log {\big (}\operatorname {rad} (abc){\big )}{\Big )}^{\omega }}{\omega !}}}$

with κ an absolute constant. After some computational experiments he found that a value of ${\displaystyle 6/5}$  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

${\displaystyle K^{\Omega (abc)}\operatorname {rad} (abc),}$

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

${\displaystyle O{\big (}\operatorname {rad} (abc)\Theta (abc){\big )},}$

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

${\displaystyle c

holds whereas there is a constant C2 such that

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

holds infinitely often.

Browkin & Brzeziński (1994) formulated the n conjecture—a version of the abc conjecture involving n > 2 integers.

## Claimed proofs

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]

## Notes

1. ^ When a + b = c, coprimality of a, b, c implies pairwise coprimality of a, b, c. So in this case, it does not matter which concept we use.

## References

1. ^ a b c
2. ^
3. ^
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.
5. ^ 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. PMID 32246118.
6. ^ Further comment by P. Scholze at Not Even Wrong math.columbia.edu[self-published source?]
7. ^ a b c
8. ^ Bombieri (1994), p. [page needed].
9. ^
10. ^
11. ^
12. ^
13. ^
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15. ^
16. ^ The ABC-conjecture, Frits Beukers, ABC-DAY, Leiden, Utrecht University, 9 September 2005.
17. ^ Mollin (2009); Mollin (2010, p. 297)
18. ^
19. ^ Granville, Andrew; Tucker, Thomas (2002). "It's As Easy As abc" (PDF). Notices of the AMS. 49 (10): 1224–1231.
20. ^ 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
21. ^ "Synthese resultaten", RekenMeeMetABC.nl (in Dutch), archived from the original on December 22, 2008, retrieved October 3, 2012.
22. ^ "Data collected sofar", ABC@Home, archived from the original on May 15, 2014, retrieved April 30, 2014
23. ^ "100 unbeaten triples". Reken mee met ABC. 2010-11-07.
24. ^ Bombieri & Gubler (2006), p. 404.
25. ^ "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.
26. ^ Ball, Peter (10 September 2012). "Proof claimed for deep connection between primes". Nature. doi:10.1038/nature.2012.11378. Retrieved 19 March 2018.
27. ^ 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.
28. ^ Calegari, Frank (December 17, 2017). "The ABC conjecture has (still) not been proved". Retrieved March 17, 2018.
29. ^ Revell, Timothy (September 7, 2017). "Baffling ABC maths proof now has impenetrable 300-page 'summary'". New Scientist.
30. ^ 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)
31. ^ Fesenko, Ivan (28 September 2016). "Fukugen". Inference. 2 (3). Retrieved 30 October 2021.
32. ^ Conrad, Brian (December 15, 2015). "Notes on the Oxford IUT workshop by Brian Conrad". Retrieved March 18, 2018.
33. ^ 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.
34. ^ Klarreich, Erica (September 20, 2018). "Titans of Mathematics Clash Over Epic Proof of ABC Conjecture". Quanta Magazine.
35. ^ "March 2018 Discussions on IUTeich". Retrieved October 2, 2018. Web-page by Mochizuki describing discussions and linking consequent publications and supplementary material
36. ^ 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.
37. ^ Mochizuki, Shinichi (July 2018). "Comments on the manuscript by Scholze-Stix concerning Inter-Universal Teichmüller Theory" (PDF). S2CID 174791744. Retrieved October 2, 2018.
38. ^ Mochizuki, Shinichi. "Comments on the manuscript (2018-08 version) by Scholze-Stix concerning Inter-Universal Teichmüller Theory" (PDF). Retrieved October 2, 2018.
39. ^ Mochizuki, Shinichi. "Mochizuki's proof of ABC conjecture". Retrieved July 13, 2021.

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