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.
Contents
- Formulations
- Examples of triples with small radical
- Some consequences
- Theoretical results
- Computational results
- Refined forms generalizations and related statements
- The work of Shinichi Mochizuki
- References
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".
Lucien Szpiro attempted a solution in 2007, but it was found to be incorrect. In August 2012 Shinichi Mochizuki posted his four preprints which develop a new inter-universal Teichmüller theory, with an alleged application to the proof of several famous conjectures including the abc conjecture. His papers were submitted to a mathematical journal and are being refereed, while various activities to study his theory have been run. Many mathematicians remain skeptical of his work, and it may take years for the question to be resolved due to the strangeness of his proof, and other difficulties like Mochizuki's earlier resistance to leaving Japan to explain his work to others.
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,rad(1000000) = rad(26 ⋅ 56) = 2 ⋅ 5 = 10.If a, b, and c are coprime 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: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:A third equivalent formulation of the conjecture involves the quality q(a, b, c) of the triple (a, b, c), defined as
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:
First we note that b is divisible by 9:
Using this fact we calculate:
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:
Now we claim that b is divisible by p2:
And now with a similar calculation as above we have:
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.
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:
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.
ABC@Home had found 23.8 million triples.
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
This incited 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
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:
Baker (1998) also describes related conjectures of Andrew Granville that would give upper bounds on c of the form
where Ω(n) is the total number of prime factors of n and
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
holds whereas there is a constant C2 such that
holds infinitely often.
Browkin & Brzeziński (1994) formulated the n conjecture—a version of the abc conjecture involving n > 2 integers.
The work of Shinichi Mochizuki
In August 2012, Shinichi Mochizuki released a series of four preprints on Inter-universal Teichmuller Theory which is then applied to prove several famous conjectures in number theory, including Szpiro's conjecture, the hyperbolic Vojta's conjecture and the abc conjecture. Mochizuki calls the theory on which this proof is based "inter-universal Teichmüller theory (IUT)". The theory is radically different from any standard theories and goes well outside the scope of arithmetic geometry. It was developed over two decades with the last four IUT papers occupying the space of over 500 pages and using many of his prior published papers.
An error in the last of the articles was pointed out by Vesselin Dimitrov and Akshay Venkatesh in October 2012, and Mochizuki revised appropriate parts of his papers on "inter-universal Teichmüller theory". Mochizuki has refused all requests for media interviews, but released progress reports in December 2013 and December 2014. He has invested hundreds of hours to run seminars and meetings to discuss his theory. According to Mochizuki, verification of the core proof is "for all practical purposes, complete." However, he also stated that an official declaration should not 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.
The first international workshop on Mochizuki's theory was organized by Ivan Fesenko and held in Oxford in December 2015. It helped to increase the number of mathematicians who had thoroughly studied parts of the IUT papers or related prerequisite papers. The next workshop on IUT Summit was held at the Research Institute for Mathematical Sciences in Kyoto in July 2016.
There are several introductory texts and surveys of the theory, written by Mochizuki and other mathematicians.