The abc conjecture

Recently, the Japanese mathematician Shinichi Mochizuki announced that he had a proof of the abc conjecture. This claim was significant enough to receive plenty of major news coverage, including from the New York Times. His strategy was to use entirely novel self-developed techniques — so-called inter-universal Teichmüller theory. In fact, practically no one else in the mathematics community is conversant in these techniques, so it will take quite a long time before the proof is sufficiently refereed. If it turns out to be accurate, however, it will undoubtedly a huge achievement!

So what is the abc conjecture, anyway? Essentially, it lies somewhere in the intersection of additive and multiplicative number theory. Curiously, it’s a statement about one of the simplest equations possible:

\( a + b = c \)

for natural numbers \( a \), \(b\), and \(c\). Naturally, there are infinitely many solutions to this equation. Now, it’s obvious that this equation is additive, but we can give it a multiplicative flavor if we consider the various prime divisors of \(a\), \(b\), and \(c\). In particular, for this conjecture, we’re interested in repeated divisors — cases where \(a\), \(b\), and \(c\) are divisible by perfect powers.

It’s probably helpful to use a concrete example to illustrate some of the ideas here. So suppose \( a = 81, b = 128, c = 209. \) We see that indeed \( a + b = c. \) Now, \( a = 3^4, b = 2^7, \) but \( c = 11 \cdot 19. \) Thus, although \( a \) and \( b \) are divisible by large prime powers, it seems that \( c \) is not. Of course this is simply one example, but in general this sort of pattern appears to be true. Try it out for yourself and see if you can find any counterexamples!

In order to describe the abc conjecture precisely, it will help to introduce a little terminology. Let’s define the radical of a natural number \( n \), denoted \( \mathrm{rad}(n), \)  to be the product of the distinct prime factors of \( n. \) For instance,

\( \mathrm{rad}(12) = \mathrm{rad}(72) = 2 \cdot 3 = 6, \mathrm{rad}(2^7) = 2, \mathrm{rad}(1) = 1, \) etc.

One formulation of the abc conjecture tries to compare \( c \) to \( \mathrm{rad} (abc) \). We can think of this in the following way: to what power must we raise the radical of \( abc \) in order to obtain \( c \)? In other words, in the form of an equation, what exponent \( q \) satisfies

\( \displaystyle (\mathrm{rad} (abc))^{q} = c \)?

The more that \( c \) is highly divisible by primes, the larger value of \( q \) we will need. Solving for \( q \), we obtain

\( \displaystyle q = \frac{\log c}{\log (\mathrm{rad} (abc))} \).

For convenience, let’s call this quantity \( q \) the quality of the abc-solution. We claim that large values of \( q \) arise when \( a\), \(b\), and \(c\) are divisible by prime powers. It’s worth verifying this claim in order to get some intuition for how this quantity \( q \) is a useful construct. So suppose for the sake of argument that \(a\), \(b\), and \(c\) are all perfect powers. For instance, let \( a = x^n, b = y^n, c = z^n \) for some natural numbers \( x, y, z, n. \) Then we get the infamous Fermat equation \( x^n + y^n = z^n \). Of course, it’s now known that no solutions exist for \( n \ge 3 \), but for the sake of argument, let’s suppose solutions do in fact exist for arbitrarily large values of \( n \). What is the quality of such an abc-solution? Well,

\(\begin{align*} \log c &= \log (\max(a, b, c))\\
&\ge \frac{1}{3}(\log a + \log b + \log c)\\
&= \frac{1}{3}\log abc\\
&= \frac{1}{3}\log (x^ny^nz^n)\\
&= \frac{1}{3}\log(xyz)^n\\
&= \frac{n}{3}\log(xyz)\\
&\ge \frac{n}{3}\log(\mathrm{rad}(abc)).


\( \displaystyle q =\frac{\log c}{\log (\mathrm{rad} (abc))} \ge \frac{n}{3}. \)

If \( n \) can be arbitrarily large, then so can \( q. \) However, the abc conjecture establishes a restriction on the size of \( q \) as follows.
The abc conjecture: For any real number \(C \) greater than \(1\), all but a finite number of \(abc\)-solutions have a quality \(q\) less than or equal to \(C\).
There are several alternative formulations as well — you can check them out at Wikipedia or MathWorld — but this one, adapted from Barry Mazur’s piece Questions about Number, appealed to me most. Let’s hope that Mochizuki’s proof is successful and opens up new doors in number theory! For further reading and research, here is a list of the consequences of the abc conjecture.

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