Abc conjecture

The abc conjecture (also known as the Oesterlé–Masser conjecture) is a big idea in number theory. It was talked about by two mathematicians, Joseph Oesterlé and David Masser, way back in 1985. The conjecture talks about three whole numbers, a, b, and c, that have no common factors besides 1 and add up to each other (like a + b = c).

The conjecture says something interesting about the prime numbers that make up a, b, and c. It suggests that the product of these prime factors can't be too small compared to c. If this conjecture is true, it would help solve many other big questions in number theory!

Many mathematicians have tried to prove the abc conjecture, but it’s still not settled. One mathematician, Shinichi Mochizuki, said he had a proof in 2012, but most mathematicians still think the conjecture isn’t proven yet. According to mathematician Dorian Goldfeld, it might be “The most important unsolved problem in Diophantine analysis.”

Formulations

The abc conjecture talks about three numbers, a, b, and c, that add up to each other (a + b = c) and have no common factors besides 1. The conjecture suggests that the product of the different prime numbers that make up a, b, and c together can't be much smaller than c itself very often.

One way to think about it is by using a special math term called the "radical" of a number, which is just the product of its different prime factors. For most triples of numbers a, b, and c that add up, c will be bigger than the radical of abc raised to a power just a bit more than 1. There are some special triples where this isn't true, but the conjecture says these are rare and there should be only a few of them.

Examples of triples with small radical

The abc conjecture talks about three numbers, a, b, and c, that add up to each other (a + b = c). It suggests that the product of the different prime numbers that can divide a, b, and c together cannot be much smaller than c itself very often.

To see why this is interesting, we can look at some special examples. For instance, if we pick a = 1, b = 2⁶ⁿ − 1, and c = 2⁶ⁿ (where n is greater than 1), we find that b is divisible by 9. This helps us calculate the "radical" of abc — a way to measure the product of different prime factors — and shows it can be surprisingly small compared to c. Similar examples exist using other numbers, helping mathematicians explore how often such small radicals occur.

Main article: Fermat's little theorem

Some consequences

The abc conjecture has many important results and ideas that follow from it. It connects to several big questions in math, like Roth's theorem about approximating numbers and the Mordell conjecture, which was already proven.

If the abc conjecture is true, it would help solve other problems too, such as Fermat's Last Theorem, which says you can't write a number as the sum of two powers that equal another power, for exponents bigger than two. The abc conjecture would also give insights into other guesses about numbers, like the Erdős–Woods conjecture and the Beal conjecture.

Theoretical results

The abc conjecture suggests that for three numbers a, b, and c that add up to each other (a + b = c), the product of their prime factors can't be much smaller than c. Scientists have found some mathematical limits that support this idea. They showed that c can be kept within certain ranges based on the prime factors of a, b, and c. These ranges help us understand how close the abc conjecture might be to true.

Researchers also discovered that there are many sets of numbers (a, b, c) where c is larger than a special calculation involving the prime factors of a, b, and c. This gives more evidence that the abc conjecture might hold true in many cases. A project called ABC@Home has found millions of such number sets, helping scientists study the conjecture further.

Refined forms, generalizations and related statements

The abc conjecture is related to a theorem about polynomials called the Mason–Stothers theorem.

Some mathematicians have suggested even stronger versions of the abc conjecture. For example, one idea is to make the rule about the prime factors of a, b, and c a bit stricter.

Other mathematicians have created new guesses inspired by the abc conjecture, such as a version that involves more than three numbers instead of just three.

Claimed proofs

In 2007, a mathematician named Lucien Szpiro tried to solve the abc conjecture, but his solution was later found to be incorrect.

Starting in 2012, another mathematician, Shinichi Mochizuki, claimed to have proven the abc conjecture using a new theory called inter-universal Teichmüller theory. However, many other mathematicians found his work very hard to understand and believed there were mistakes in his proof. Even after discussions and attempts to explain the theory, most mathematicians still do not accept it as a correct proof. In 2021, Mochizuki's work was published, but it continues to be viewed with skepticism by many experts.