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 c 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:
- The Fermat–Catalan conjecture: a generalisation of Fermat’s last theorem concerning powers that are sums of powers;
- The Erdős–Woods conjecture except for a finite number of counterexamples;
- The weak form of Marshall Hall’s conjecture on the separation between squares and cubes of integers;
- The Mordell conjecture (however this has already been proven in general by Gerd Faltings);
and many more.
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.