Abc conjecture

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: c > rad ⁡ ( a b c ) 1 + ε .

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: c < K ε ⋅ rad ⁡ ( a b c ) 1 + ε .

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 ) = log ⁡ ( c ) log ⁡ ( rad ⁡ ( a b c ) ) .

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 , b = 2 6 n − 1 , c = 2 6 n , n > 1.

First we note that b is divisible by 9:

b = 2 6 n − 1 = 64 n − 1 = ( 64 − 1 ) ( ⋯ ) = 9 ⋅ 7 ⋅ ( ⋯ )

Using this fact we calculate:

rad ⁡ ( a b c ) = rad ⁡ ( a ) rad ⁡ ( b ) rad ⁡ ( c ) = rad ⁡ ( 1 ) rad ⁡ ( 2 6 n − 1 ) rad ⁡ ( 2 6 n ) = 2 rad ⁡ ( 2 6 n − 1 ) = 2 rad ⁡ ( 9 ⋅ b 9 ) ⩽ 2 ⋅ 3 ⋅ b 9 = 2 b 3 < 2 3 c b < c

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 , b = 2 p ( p − 1 ) n − 1 , c = 2 p ( p − 1 ) n , n > 1.

Now we claim that b is divisible by p²:

b = 2 p ( p − 1 ) n − 1 = ( 2 p ( p − 1 ) ) n − 1 = ( 2 p ( p − 1 ) − 1 ) ( ⋯ ) = p 2 ⋅ r ( ⋯ ) p 2 | 2 p ( p − 1 ) − 1

And now with a similar calculation as above we have:

rad ⁡ ( a b c ) < 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 (in dimension 1). (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.

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.

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)

Fermat's Last Theorem has a famously difficult proof by Andrew Wiles. However it follows easily, at least for n ≥ 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 n ≥ 6 .

The Beal conjecture, a generalization of Fermat's last theorem proposing that if A, B, C, x, y, and z are positive integers with A^x + B^y = C^z and x, y, z > 2, then A, B, and C have a common prime factor.

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 ⁡ ( K 1 rad ⁡ ( a b c ) 15 ) (Stewart & Tijdeman 1986), c < exp ⁡ ( K 2 rad ⁡ ( a b c ) 2 3 + ε ) (Stewart & Yu 1991), and c < exp ⁡ ( K 3 rad ⁡ ( a b c ) 1 3 + ε ) (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.

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 ( ε − ω rad ⁡ ( a b c ) ) 1 + ε over ε > 0 occurs when ε = ω log ⁡ ( rad ⁡ ( a b c ) ) .

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

c < κ rad ⁡ ( a b c ) ( log ⁡ ( rad ⁡ ( a b c ) ) ) ω ω !

with κ an absolute constant. After some computational experiments he found that a value of 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 < ( rad ⁡ ( a b c ) ) 1 + 3 4 .

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

K Ω ( a b c ) r a d ( a b c ) ,

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

O ( r a d ( a b c ) Θ ( 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 ⁡ ( 4 3 log ⁡ k log ⁡ log ⁡ k ( 1 + log ⁡ log ⁡ log ⁡ k 2 log ⁡ log ⁡ k + C 1 log ⁡ log ⁡ k ) )

holds whereas there is a constant C₂ such that

c > k exp ⁡ ( 4 3 log ⁡ k log ⁡ log ⁡ k ( 1 + log ⁡ log ⁡ log ⁡ k 2 log ⁡ log ⁡ k + C 2 log ⁡ log ⁡ k ) )

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.