THE BLOG
07/31/2013 11:57 am ET Updated Sep 30, 2013

# What Is the ABC Conjecture?

This post may be a little late but I saw a book on the ABC Conjecture the other day when I was scanning through the bookstore with a friend of mine who asked me to explain what the conjecture is about. The ABC Conjecture got some media attention when Professor Shinichi Mochizuki published a possible proof for the conjecture last August who researches at the university where I am currently currently spending a semester at so I thought it might be nice to do a post to explain what the conjecture states.

Before anything, conjectures are proposals which are thought to be true but has not been proven yet. When a conjecture is proved, it becomes a theorem. A counterexample can also be found to show that the conjecture is NOT correct.

This post does not contain any historical information about the conjecture nor anything on Professor Mochizuki's proof (it is 500 pages long and it is built upon his previous papers).

The ABC Conjecture proposed in the 1980s by Oesterlé and Masser is stated as such:

"For any infinitesimal $\epsilon>0}$, there exists a constant $C_{\epsilon }$ such that for any three relatively prime integers $a$, $b$, $c$ satisfying

$a+b=c$

the inequality

$\textup{max}(|a|,|b|,|c|))\leq C_{\epsilon }\prod_{p|abc} p^{1+\epsilon }$

holds".

This was taken straight out of Wolfram MathWorld which I believe to be a reliable source for information on mathematics.

Let's see what the set up of the conjecture is - the ingredients which make up the problem.

An "infinitesimal" number bigger than zero can be thought as the smallest number you can come up with and make that number even smaller by dividing it by the biggest number you can come up with. So the number is very, very (infinitely) tiny but the important fact is that it is bigger than 0. We will call that number $\epsilon$ which is a common notation in mathematics for an infinitesimally small number.

A constant $C_{\epsilon }$ is just a number but its size depends on how big $\epsilon$ is, hence the subscript. This value is assumed to be bigger than 0.

Integers are numbers that can be written without fractions or decimal points. Examples are -4, 0, 100000.

Additionally, those prime integers have to be "pairwise coprime". Two integers are said to be "coprime" if the only integer which divides those two integers is 1. Pairwise means {a, b}, {a, c}, and {b, c} all satisfy the coprimeness.
If you can recall something from your mathematics classes, you may remember that any number can be uniquely (written only in one way) decomposed as a product of prime numbers, in other words, you can always rewrite a number as products of prime numbers. For example: $1200 = 2^4\times 3\times 5^2$.
So to check that all those integers are coprime, you can decompose a, b, and c into its primes and make sure that the primes used to build one number is not used to build the others.

For example: Let $a=1024$ and $b=81$.
Then $c=1105 (=a+b=1024+81)$.
Decomposition: $a=1024=2^{10}$, $b=81=3^4$, $c=1105=5\times 13\times 17$

We have that a, b, and c are pairwise coprime since we don't have any repeated primes.

To foreshadow the importance of this conjecture, prime numbers in mathematics are like atoms in physics, they are the fundamental building blocks of all numbers. Learning about prime numbers will allow us to learn more about the fundamentality of numbers in general.

Now that we have all the ingredients for the conjecture, let's break the equations down to see what each component means.

On the left hand side, we have $\textup{max}(|a|,|b|,|c|))$.

This means the left hand side will be the absolute value of one of the integer a, b, or c with the biggest absolute number. An easier way to see which value this takes is by rearranging the equation $a+b=c$ so that there are no negative numbers on either sides of the equality and the value will equal the side with just one component.

For example, $2-3=-1$ can be rewritten as $2+1=3$ and as you can imagine, the biggest number will lie on the side with only one number - this is the value of $\textup{max}(|a|,|b|,|c|))$ which in this case is 3.

Now for the scariest looking part of the equation, $\prod_{p|abc} p^{1+\epsilon }$.

Let's ignore the power $1+\epsilon$ for now.
This is called the "square free part" of the product $abc$ and is also called the "radical" of $abc$ denoted as $rad(abc)$.

$rad(abc)$ is the product of all the prime numbers which built up a, b, and c (without the duplicates). Using the above example, since $a=1024=2^{10}$, $b=81=3^4$, and $c=1105=5\times 13\times 17$, we have:

$rad(abc)=2\times 3\times 5\times 13\times 17=6630$

Now back to powering it by $1+\epsilon$. This means to take the $rad(abc)$ and make it a tiny tiny bit bigger since anything to the power of a number bigger than 1 will be bigger than its original value. But remember that $\epsilon$ was very small, so it's only bigger by a minuscule bit.

Now we know what both sides of the inequality are, let's put the pieces altogether. I'll restate the theorem again for reference:

"For any infinitesimal $\epsilon>0}$, there exists a constant $C_{\epsilon }$ such that for any three relatively prime integers $a$, $b$, $c$ satisfying

$a+b=c$

the inequality

$\textup{max}(|a|,|b|,|c|))\leq C_{\epsilon }\prod_{p|abc} p^{1+\epsilon }$

holds".

What this says is that if we make $rad(abc)$ a little bigger, then we can find an upper bound to how big $\textup{max}(|a|,|b|,|c|))$ can be. That upper bound is the product with $C_{\epsilon }$.
Conversely, without the power of $1+\epsilon$, there is an infinite number of {a, b, c} which exceeds $rad(abc)$, we cannot find a $C_{\epsilon }$ that will bound the left hand side.

In other words, when we have $rad(abc)$ to the power of $1+\epsilon$, only a finite number of the triples {a, b, c} are bigger than that value.

So how can one appreciate this conjecture?

If this conjecture is proven to be correct, then few other open problems can also be proven, one of which is the Fermat's Last Theorem (although this is no longer an open problem). For a while, mathematicians thought that this was the way to solve Fermat's Last Theorem although it did not turn out to be that way.

This may still be unsatisfactory for many non-mathematicians.

As I stated above, the Fundamental Theorem of Arithmetics states that every number can be decomposed into products of prime numbers. This means that prime numbers play the same role as atom play in physics but in mathematics. Learning more about atoms do help us understand more about the world at the fundamental levels. Similarly, learning about prime numbers allow us to learn more about the fundamental building blocks of all the numbers. ABC Conjecture is one of the features that mathematicians believe prime numbers have.

Arguably, knowing about prime numbers is not going to be particularly helpful with the numbers that we deal with everyday -- just like how knowledge about atoms is not very practical at a macroscopic level.