abc conjecture

The conjecture essentially states that the product of the distinct prime factors of

Mathematician Dorian Goldfeld described the abc conjecture as "The most important unsolved problem in Diophantine analysis".

[1] 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.

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.

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).

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 2p(p−1) − 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 (Lando & Zvonkin 2004, p. 137) for The abc conjecture has a large number of consequences.

There are also theoretical results that provide a lower bound on the best possible form of the abc conjecture.

In particular, Stewart & Tijdeman (1986) showed that there are infinitely many triples (a, b, c) of coprime integers with a + b = c and for all k < 4.

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.

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 where ω is the total number of distinct primes dividing a, b and c.[27] Andrew Granville noticed that the minimum of the function

This inspired Baker (2004) to propose a sharper form of the abc conjecture, namely: with κ an absolute constant.

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 a more precise inequality based on Robert & Tenenbaum (2013).

Browkin & Brzeziński (1994) formulated the n conjecture—a version of the abc conjecture involving n > 2 integers.

Lucien Szpiro proposed a solution in 2007, but it was found to be incorrect shortly afterwards.

[5] He released a series of four preprints developing a new theory he called inter-universal Teichmüller theory (IUTT), which is then applied to prove the abc conjecture.

[29] The papers have not been widely accepted by the mathematical community as providing a proof of abc.

[30] This is not only because of their length and the difficulty of understanding them,[31] but also because at least one specific point in the argument has been identified as a gap by some other experts.

[32] Although a few mathematicians have vouched for the correctness of the proof[33] and have attempted to communicate their understanding via workshops on IUTT, they have failed to convince the number theory community at large.

[34][35] In March 2018, Peter Scholze and Jakob Stix visited Kyoto for discussions with Mochizuki.

Scholze and Stix wrote a report asserting and explaining an error in the logic of the proof and claiming that the resulting gap was "so severe that ... small modifications will not rescue the proof strategy";[32] Mochizuki claimed that they misunderstood vital aspects of the theory and made invalid simplifications.

Mochizuki is chief editor of the journal but recused himself from the review of the paper.

[6] The announcement was received with skepticism by Kiran Kedlaya and Edward Frenkel, as well as being described by Nature as "unlikely to move many researchers over to Mochizuki's camp".

Mathematician Joseph Oesterlé
Mathematician David Masser