The abc conjecture (also known as Oesterlé–Masser conjecture) is a conjecture in number theory, first proposed by Joseph Oesterlé and David Masser in 1985. The conjecture is stated in terms of three positive integers, a, b and c (whence comes the name), which have no common factor and satisfy a + b = c. If d denotes the product of the distinct prime factors of abc, the conjecture essentially states that d is rarely much smaller than c.
Although there is no obvious strategy for resolving the problem, it has already become well known for the number of interesting consequences it entails. Dorian M. Goldfeld described the abc conjecture as "the most important unsolved problem in Diophantine analysis".[1]
Are there for every ε > 0, only finitely many triples of coprime positive integers such that where d denotes the product of the distinct prime factors of abc? |
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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[2] 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 ε>0 there exist only finitely many triples (a,b,c) of positive coprime integers with a + b = c such that
An equivalent formulation states that for any ε > 0, there exists a constant K such that, for all triples of coprime positive integers (a, b, c) satisfying a + b = c, the inequality
holds.
A third formulation of the conjecture involves the quality q(a, b, c) of the triple (a, b, c), defined by:
For example
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 abc conjecture states that, for any 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.
The condition that ε > 0 is necessary for the truth of the conjecture, as there exist infinitely many triples a, b, c with rad(abc) < c. For instance, such a triple may be taken as
As a and c together contribute only a factor of two to the radical, while b is divisible by 9, rad(abc) < 2c/3 for these examples. By replacing the exponent 6n by other exponents forcing b to have larger square factors, the ratio between the radical and c may be made arbitrarily large. Another triple with a particularly small radical was found by Eric Reyssat:[3]
The conjecture has not been proven, but it has a large number of interesting consequences. These include both known results, and conjectures for which it gives a conditional proof.
While the first group of these have now been proven, the abc conjecture itself remains of interest, because of its numerous links with deep questions in number theory.
It remains unknown whether c can be upper bounded by a near-linear function of the radical of abc, as the abc conjecture states, or even whether it can be bounded by a polynomial of rad(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.
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.
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 | 14.482.059 | 2.352.105 | 449.194 | 24.013 | 1.843 | 160 |
As of March 2011[update], ABC@Home has found 20.9 million triples, and its present goal is to obtain a complete list of all ABC triples (a,b,c) with c no more than 1020.[15]
q | a | b | c | Discovered by | |
---|---|---|---|---|---|
1 | 1.6299 | 2 | 310109 | 235 | Eric Reyssat |
2 | 1.6260 | 112 | 325673 | 22123 | Benne de Weger |
3 | 1.6235 | 19·1307 | 7·292318 | 2832254 | Jerzy Browkin, Juliusz Brzezinski |
4 | 1.5444 | 724123113 | 111613279 | 2·33523953 | Abderrahmane Nitaj |
5 | 1.4805 | 52279·45949 | 321318613 | 223174251217333 | Frank Rubin |
where the quality q(a, b, c) of the triple (a, b, c), defined by:
A stronger inequality proposed in 1996 by Alan Baker states that in the inequality, one can replace rad(abc) by
where ω is the total number of distinct primes dividing a, b and c. A related conjecture of Andrew Granville states that on the RHS we could also put
where Θ(n) is the number of integers up to n divisible only by primes dividing n.
In 1994, Jerzy Browkin and Juliusz Brzeziński formulated the n-conjecture[17]—a version of the abc conjecture involving integers.