Abc conjecture

The abc conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory that arose out of a discussion of Joseph Oesterlé and David Masser in 1985. It is stated in terms of three positive integers ' and ' (hence the name) that are relatively prime and satisfy '. The conjecture essentially states that the product of the distinct prime factors of ' cannot often be much smaller than '. A number of famous conjectures and theorems in number theory would follow immediately from the abc conjecture or its versions. Mathematician Dorian Goldfeld described the abc conjecture as "The most important unsolved problem in Diophantine analysis".

The abc conjecture originated as the outcome of attempts by Oesterlé and Masser to understand the Szpiro conjecture about elliptic curves, which involves more geometric structures in its statement than the abc conjecture. The abc conjecture was shown to be equivalent to the modified Szpiro's conjecture.

Various attempts to prove the abc conjecture have been made, but none have gained broad acceptance. Shinichi Mochizuki claimed to have a proof in 2012, but the conjecture is still regarded as unproven by the mainstream mathematical community.

Formulations

Before stating the conjecture, the notion of the radical of an integer must be introduced: for a positive integer ', the radical of ', denoted ', is the product of the distinct prime factors of '. For example,

If a, b, and c are coprime positive integers such that a + b = c, it turns out that "usually" '. The abc conjecture deals with the exceptions. Specifically, it states that:

An equivalent formulation is:

Equivalently (using the little o notation):

A fourth equivalent formulation of the conjecture involves the quality q(a, b, c) of the triple (a, b, c), which is defined as

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 fourth formulation is:

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) that 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 c > rad(abc). For example, let

The integer b is divisible by 9:

Using this fact, the following calculation is made:

By replacing the exponent 6n with 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 it may be plausibly claimed that b is divisible by p²:

The last step uses the fact that p² divides 2^{p(p−1)} − 1. This follows from Fermat's little theorem, which shows that, for p > 2, 2^{p−1} = pk + 1 for some integer k. Raising both sides to the power of p then shows that 2^{p(p−1)} = p²(...) + 1.

And now with a similar calculation as above, the following results:

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 for

Some consequences

The abc conjecture has a large number of consequences. These include both known results (some of which have been proven separately only since the conjecture has been stated) and conjectures for which it gives a conditional proof. The consequences include:

• Roth's theorem on Diophantine approximation of algebraic numbers.
• The Mordell conjecture (already proven in general by Gerd Faltings).
• As equivalent, Vojta's conjecture in dimension 1.
• The Erdős–Woods conjecture allowing for a finite number of counterexamples.
• The existence of infinitely many non-Wieferich primes in every base b > 1.
• The weak form of Marshall Hall's conjecture on the separation between squares and cubes of integers.
• Fermat's Last Theorem has a famously difficult proof by Andrew Wiles. However it follows easily, at least for , 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 .
• The Fermat–Catalan conjecture, a generalization of Fermat's Last Theorem concerning powers that are sums of powers.
• The L-function L(s, χ_d) formed with the Legendre symbol, has no Siegel zero, given a uniform version of the abc conjecture in number fields, not just the abc conjecture as formulated above for rational integers.
• A polynomial P(x) has only finitely many perfect powers for all integers x if P has 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.
• As equivalent, 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