abc Conjecture - Formulations

Formulations

The abc conjecture can be expressed as follows: For every ε > 0, there are only finitely many triples of coprime positive integers a + b = c such that c > d (1+ε), where d denotes the product of the distinct prime factors of abc.

To illustrate the terms used, if
a = 16 = 24,
b = 17, and
c = 16 + 17 = 33 = 3·11,
then d = 2·17·3·11 = 1122, which is greater than c. Therefore, for all ε > 0, c is not greater than d(1+ε). According to the conjecture, most coprime triples where a + b = c are like the ones used in this example, and for only a few exceptions is c > d(1+ε).

Adding additional terminology: 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 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 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, 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,

• 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.

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.