abc Conjecture

What is it?

The abc conjecture was first posed by Joseph Oesterlé in 1985 and David Masser in 1988. The conjecture begins by presenting 3 distinct positive integers a, b and that are relatively prime and satisfy the equation a + b = c. It states that, if the number d  is the product of the distinct prime numbers of abc, then d is usually much larger smaller than c. This product is defined mathematically as the ‘radical’ of abc and hence the conjecture can be expressed more formally as:

If a, b, and c are coprime positive integers such that a + b =c, it turns out that “usually” c < rad(abc).

There are a few exceptions to this. To deal with this, the abc conjecture specifically states that:

For every ε > 0, there exist only finitely many triples (a, b, c) of coprime positive integers, with a +b = c, such that:

Consequences if it is TRUE

The abc conjecture is linked to many other questions in number theory; if it were to be true, then it would imply that many other conjectures are also true, including:

and many more.

Proof

In 2012, the Japanese mathematician Shinchi Mochizuki published a 500 page proof of the abc conjecture. However, it was so complex that no mathematicians could understand it due to the fact that it used a new mathematical framework known as inter-universal Teichmüller Theory. In a verification report published on December 2014, Mochizuki stated that

“With the exception of the handful of researchers already involved in the verification activities concerning IUTeich (inter-universal Teichmüller Theory) discussed in the present report, every researcher in arithmetic geometry throughout the world is a complete novice with respect to the mathematics surrounding IUTeich, and hence, in particular, is simply not qualified to issue a definitive (i.e., mathematically meaningful) judgment concerning the validity of IUTeich on the basis of a ‘deep understanding’ arising from his/her previous research achievements.”

However, a few weeks ago, four dozen mathematicians converged to hear Mochizuki present his own work at a conference in Kyoto University. This was a ‘breakthrough’ as described by Ivan Fesenko: “It was the key part of the meeting… He was climbing the summit of his theory, and pulling other participants with him, holding their hands.”

Now, around 10 mathematicians have a solid understanding of this new theory, giving a glimmer of hope for our future understanding of this proof.

Video

Sources: 1 | 2 | 3 | 4 | 5 | 6

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