# abc conjecture

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. The precise statement is given below.

The abc conjecture has already become well known for the number of interesting consequences it entails. Many famous conjectures and theorems in number theory would follow immediately from the abc conjecture. Goldfeld (1996) described the abc conjecture as "the most important unsolved problem in Diophantine analysis".

Several solutions have been proposed to the abc conjecture, the most recent of which is still being evaluated by the mathematical community.

## Formulations

Before we state the conjecture we need to 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

If a, b, and c are coprime[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 ε > 0, there exist only finitely many triples (a, b, c) of coprime positive integers, with a + b = c, such that:
$c>\operatorname{rad}(abc)^{1+\varepsilon}.$

An equivalent formulation states that:

ABC Conjecture II. For every ε > 0, there exists a constant Kε such that for all triples (a, b, c) of coprime positive integers, with a + b = c, such that:
$c < K_{\varepsilon} \cdot \operatorname{rad}(abc)^{1+\varepsilon}.$

A third equivalent formulation of the conjecture involves the quality q(a, b, c) of the triple (a, b, c), defined as

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

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 ε > 0, there exist only finitely many triples (a, b, c) of coprime positive integers with a + b = c such that q(a, b, c) > 1 + ε.

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) which 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 rad(abc) < c. For example let:

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

First we note that b is divisible by 9:

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

Using this fact we calculate:

\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 && b< c \end{align}

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:

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

Now we claim that b is divisible by p2:

\begin{align} b &= 2^{p(p-1)n} -1 \\ &= \left (2^{p(p-1)} \right )^n - 1^n \\ &= \left ( 2^{p(p-1)} - 1 \right ) (\cdots) \\ &= p^2 \cdot r (\cdots) && p^2 | 2^{p(p-1)} - 1 \end{align}

And now with a similar calculation as above we have:

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

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 = 6,436,341,
c = 235 = 6,436,343,

## Some consequences

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.

$c_f = \prod_{\text{prime }p} x_i \left ( 1 - \frac{\omega\,\!_f (p)}{p^{2+q_p}} \right ).$
• Andrew Wiles' proof Fermat's Last Theorem is famous for its difficulty. However if a strong effective form of the abc conjecture is correct, then one can give a very short and easy proof of Fermat's Last theorem.[5]

## Theoretical results

The abc conjecture implies that c can be bounded above by a near-linear function of the radical of abc. However, exponential bounds are known. Specifically, the following bounds have been proven:

$c < \exp{ \left(K_1 \operatorname{rad}(abc)^{15}\right) }$ (Stewart & Tijdeman 1986),
$c < \exp{ \left(K_2 \operatorname{rad}(abc)^{\frac{2}{3} + \varepsilon}\right) }$ (Stewart & Yu 1991), and
$c < \exp{ \left(K_3 \operatorname{rad}(abc)^{\frac{1}{3} + \varepsilon}\right) }$ (Stewart & Yu 2001).

In these bounds, K1 is a constant that does not depend on a, b, or c, and K2 and K3 are constants that depend on ε (in an effectively computable way) but not on a, b, or c. The bounds apply to any triple for which c > 2.

## 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[6]
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

ABC@Home had found 23.8 million triples.[7]

Highest quality triples[8]
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 (Bombieri & Gubler 2006, p. 404).

Andrew Granville noticed that the minimum of the function

$({\varepsilon}^{-\omega}\operatorname{rad}(abc))^{1+\varepsilon}$

over ε > 0 occurs when

${\varepsilon}= \frac{\omega}{\log(\operatorname{rad}(abc))}.$

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

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

with κ an absolute constant. After some computational experiments he found that a value of $\tfrac{6}{5}$ was admissible for κ.

This version is called "explicit abc conjecture".

From the previous inequality, Baker deduced a stronger form of the original abc conjecture:

Baker's Strong Form of ABC Conjecture. Let a, b, c be coprime positive integers with a + b = c; then:
$c<(\operatorname{rad}(abc))^{1+\frac{3}{4}}$.

Baker (1998) also describes related conjectures of Andrew Granville that would give upper bounds on c of the form

$K^{\Omega(a b c)} \mathrm{rad}(a b c),$

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

$O(\mathrm{rad}(a b c) \Theta(a b c)),$

where Θ(n) is the number of integers up to n divisible only by primes dividing n.

Robert, Stewart & Tenenbaum (2014) proposed more precise inequality based on Robert & Tenenbaum (2013). Let k = rad(abc). They conjectured there is a constant C1 such that

$c

holds whereas there is a constant C2 such that

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

holds infinitely often.

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

## Attempts at solution

Lucien Szpiro attempted a solution in 2007, but it was found to be incorrect.[9]

In August 2012, Shinichi Mochizuki released a series of four preprints containing a claim to a proof of the abc conjecture.[10] Mochizuki calls the theory on which this proof is based "inter-universal Teichmüller theory (IUT)", and it has other applications, including a proof of Szpiro's conjecture and Vojta's conjecture.[11] Experts were expected to take months to check Mochizuki's new mathematical machinery, which was developed over decades in 500 pages of preprints and several of his prior papers.[12]

When an error in one of the articles was pointed out by Vesselin Dimitrov and Akshay Venkatesh in October 2012, Mochizuki posted a comment on his website acknowledging the mistake, stating that it would not affect the result, and promising a corrected version in the near future.[13] He revised all of his papers on "inter-universal Teichmüller theory", the latest of which is dated September 2015.[10] Mochizuki has refused all requests for media interviews, but released progress reports in December 2013[14] and December 2014.[15] According to Mochizuki, verification of the core proof is "for all practical purposes, complete." However, he also stated that an official declaration shouldn't happen until some time later in the 2010s, due to the importance of the results and new techniques. In addition, he predicts that there are no proofs of the abc conjecture that use significantly different techniques than those used in his papers.[15] There was a workshop on IUT at Kyoto University in March 2015, another one was held at Clay Mathematics Institute in December 2015 [16] and a third one will be held at the Research Institute for Mathematical Sciences in Kyoto in July 2016.[17]

Fesenko has estimated that it would take an expert in arithmetic geometry some 500 hours to understand his work. So far, only four mathematicians say that they have been able to read the entire proof.[18]

## Notes

1. ^ 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.
2. ^ http://www.math.uu.nl/people/beukers/ABCpresentation.pdf
3. ^ Mollin (2009)
4. ^ Mollin (2010) p. 297
5. ^ Granville, Andrew; Tucker, Thomas (2002). "It’s As Easy As abc". Notices of the AMS 49 (10): 1224–1231.
6. ^ "Synthese resultaten", RekenMeeMetABC.nl (in Dutch), retrieved October 3, 2012.
7. ^ "Data collected sofar", ABC@Home, retrieved April 30, 2014
8. ^ "100 unbeaten triples". Reken mee met ABC. 2010-11-07.
9. ^ "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.
10. ^ a b Mochizuki, Shinichi (May 2015). Inter-universal Teichmuller Theory I: Construction of Hodge Theaters, Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic Evaluation, Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice., Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations, available at http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html
11. ^ Ball, Phillip (10 September 2012), "Proof claimed for deep connection between primes", Nature, doi:10.1038/nature.2012.11378.
12. ^ Cipra, Barry (September 12, 2012), "ABC Proof Could Be Mathematical Jackpot", News, Science.
13. ^ Kevin Hartnett (3 November 2012). "An ABC proof too tough even for mathematicians". Boston Globe.
14. ^ "On the Verification of Inter-Universal Teichmüller Theory: A Progress Report (as of December 2013)" by Shinichi Mochizuki
15. ^ a b "On the Verification of Inter-Universal Teichmüller Theory: A Progress Report (as of December 2014)" by Shinichi Mochizuki
16. ^ http://www.claymath.org/events/iut-theory-shinichi-mochizuki
17. ^ https://www.maths.nottingham.ac.uk/personal/ibf/files/kyoto.iut.html
18. ^ Castelvecchi, Davide (7 October 2015). "The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof". Nature 526. doi:10.1038/526178a.