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

rad(16) = rad(2⁴) = 2,

rad(17) = 17,

rad(18) = rad(2 ⋅ 3²) = 2 · 3 = 6.

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 K_ε 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,

rad(abc) = 15042.

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.

Thue–Siegel–Roth theorem on diophantine approximation of algebraic numbers (Bombieri 1994)
The Mordell conjecture (already proven in general by Gerd Faltings) (Elkies 1991)
It is equivalent to Vojta's conjecture. (Van Frankenhuijsen 2002)
The Erdős–Woods conjecture except for a finite number of counterexamples (Langevin 1993)
The existence of infinitely many non-Wieferich primes in every base b > 1 (Silverman 1988)
The weak form of Marshall Hall's conjecture on the separation between squares and cubes of integers (Nitaj 1996)
The Fermat–Catalan conjecture, a generalization of Fermat's last theorem concerning powers that are sums of powers (Pomerance 2008)
The L-function L(s, χ_d) formed with the Legendre symbol, has no Siegel zero (this consequence actually requires a uniform version of the abc conjecture in number fields, not only the abc conjecture as formulated above for rational integers) (Granville & Stark 2000)
P(x) has only finitely many perfect powers for integral x for P a polynomial with at least three simple zeros.^[2]
A generalization of Tijdeman's theorem concerning the number of solutions of y^m = xⁿ + k (Tijdeman's theorem answers the case k = 1), and Pillai's conjecture (1931) concerning the number of solutions of Ay^m = Bxⁿ + k.
It is equivalent to 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.^[3]^[4]
It is equivalent to the modified Szpiro conjecture, which would yield a bound of rad(abc)^1.2+ε (Oesterlé 1988).
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 ~c_fN positive integers n ≤ N for which f(n)/B' is square-free, with c_f > 0 a positive constant defined as: (Granville 1998)

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

ε^−ω rad(abc),

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 ${\varepsilon}>0$ 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: let a, b, c be coprime positive integers with a + b = c; then we have:

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<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\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

↑ 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.
↑ http://www.math.uu.nl/people/beukers/ABCpresentation.pdf
↑ Mollin (2009)
↑ Mollin (2010) p. 297
↑ Granville, Andrew; Tucker, Thomas (2002). "It’s As Easy As abc". Notices of the AMS 49 (10): 1224–1231.
↑ "Synthese resultaten", RekenMeeMetABC.nl (in Dutch), retrieved October 3, 2012 .
↑ "Data collected sofar", ABC@Home, 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 .
1 2 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
↑ Ball, Phillip (10 September 2012), "Proof claimed for deep connection between primes", Nature, doi:10.1038/nature.2012.11378 .
↑ Cipra, Barry (September 12, 2012), "ABC Proof Could Be Mathematical Jackpot", News, Science .
↑ Kevin Hartnett (3 November 2012). "An ABC proof too tough even for mathematicians". Boston Globe.
↑ "On the Verification of Inter-Universal Teichmüller Theory: A Progress Report (as of December 2013)" by Shinichi Mochizuki
1 2 "On the Verification of Inter-Universal Teichmüller Theory: A Progress Report (as of December 2014)" by Shinichi Mochizuki
↑ http://www.claymath.org/events/iut-theory-shinichi-mochizuki
↑ https://www.maths.nottingham.ac.uk/personal/ibf/files/kyoto.iut.html
↑ Castelvecchi, Davide (7 October 2015). "The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof". Nature 526. doi:10.1038/526178a.

References

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Baker, Alan (2004). "Experiments on the abc-conjecture". Publ. Math. Debrecen 65: 253–260.
Bombieri, Enrico (1994). "Roth's theorem and the abc-conjecture". preprint (ETH Zürich).
Bombieri, Enrico; Gubler, Walter (2006). Heights in Diophantine Geometry. New Mathematical Monographs 4. Cambridge University Press. doi:10.2277/0521846153. ISBN 978-0-521-71229-3. Zbl 1130.11034.
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External links

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/ABC-Conjecture.pdf
The amazing ABC conjecture
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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