Earlier this month, New Scientist reported that the journal Publications of the Research Institute for Mathematical Sciences may soon accept Shinichi Mochizuki’s articles claiming to solve the abc conjecture. Mochizuki first announced the proof of this conjecture in number theory five years ago. Since then, mathematicians have been befuddled. Some number theorists have read the proof and say it's correct, but some feel that Mochizuki has not made his arguments understandable enough for them to evaluate or adequately addressed issues they have raised.
It’s almost as if as soon as a mathematician understands the proof, they are immediately doomed never to be able to explain it to anyone else. It's a low-budget Arthurian legend: Sir Gawain and the Incommunicable Proof. What is the conjecture that's causing such a big fuss?
Number theory is famous for having easy to state but hard to prove questions. Take the twin primes conjecture, which states that there are infinitely many prime numbers that differ by only 2, like 3 and 5, or 11 and 13. Though Yitang Zhang took a big step towards proving it in 2012, showing that there are infinitely many prime numbers that differ by 70 million (the gap has since been lowered to 246), the full conjecture is still unsolved.
Fermat’s Last Theorem, one of the most famous results in mathematics, states that for whole numbers n that are greater than 2, the equation an+bn=cn has no whole number, nonzero solutions a, b, and c. Though it doesn’t take too much unraveling to understand, over 350 years elapsed between the time Fermat scrawled his claim in the margin of a book and Andrew Wiles’ successful solution to the problem.
The abc conjecture isn’t quite like those easy to state but hard to prove questions. It’s hard to prove, but it’s also hard to state. It’s not as hard to state as it is to prove, but it doesn’t roll off the tongue the way twin primes or Fermat’s Last Theorem do.
When you boil it down, the big, deep idea is of the abc conjecture is that the factors of two whole numbers have a relationship with the factors of their sum. At first glance, that’s bizarre. How would knowing something about the factors of 4 and 11 tell us about the factors of15? Would 7 and 8 tell us the same thing or a different thing? Why should multiplication and addition have anything to do with each other anyway?
To understand the abc conjecture at a more granular level, we start with two whole numbers, a and b, that share no prime factors. (A prime number is a number whose only factors are 1 and itself. For convenience, 1 itself is not considered prime.) We also look at their sum a+b=c. The triples (4,11,15) and (7,8,15) would both be allowed, but (5,10,15) and (6,9,15) would not because they share factors.
The abc conjecture is concerned with a quantity called the radical of an integer, denoted rad(n) for the number n. It’s the product of all the distinct primes that divide the integer. So the numbers 2, 4, 8, 16, and so on all have the same radical, 2, because 2 is their only prime factor. The number 60 is 22×3×5, so rad(60)=2×3×5=30.
We’ll be looking at rad(abc). If you start playing around with some examples, you’ll find that if a+b=c and a, b, and c share no factors, the number c is usually smaller than rad(abc). For example, with the numbers 4, 11, and 15, we have 15<rad(660)=330.
It doesn’t work all the time. For example, 5+27=32. 27 is a power of 3, and 32 is a power of 2, so we can compute rad(5×27×32) pretty easily. It’s 5×3×2, or 30, just squeaking under 32. For a larger example, consider 1, 4095, and 4096. 4096 is a power of 2—212, to be precise. It’s a little more work to factor 4095, but it ends up being 32×5×7×13, so rad(1×4095×4096)=2×3×5×7×13=2730.
Exceptions to the norm that c<rad(abc) are called abc-hits. There are infinitely many of them, and even some formulas for generating new ones. Instead of asking how often c is greater than rad(abc), the abc conjecture looks at how extreme the abc-hits can be and how many can be that extreme. Mathematicians measure how extreme an abc-hit is by looking at what power you’d have to raise the radical to to get c. In our example of 5, 27, and 32, we got rad(5×27×32)=30, which is less than 32, but 301.02 is greater than 32. Basically, it doesn’t take an exponent much larger than 1 to push rad(5×27×32) above 32. (1.02 isn’t the smallest power that works, but it’s an easy one to write down.) The smallest power that pushes rad(abc) over c is called the quality of the triple. That seems unnecessarily judgmental, but no one asked me.
The triple (5,27,32) has a fairly low quality, a bit under 1.02. The triple (1,8,9) has a higher quality; rad(72)=6, and you have to raise 6 to about the 1.23 power to get something bigger than 9.
900 words after starting this post, we can finally state the abc conjecture: for any number ε greater than 0, there are only finitely many triples so that c>rad(abc)1+ε. So if we took ε=0.02, for a quality of 1.02, the triple (5,27,32) does not count, but (1,8,9) does. Another way to put it is that if we set a quality threshold of any given number strictly greater than 1, only finitely many triples will exceed it.* (There are a few other equivalent formulations, but we won’t go into those here.)
We’ve gotten to the bottom of what the abc conjecture is, but its not clear yet why it is. Sure, we can ask whether for a given ε there are finitely many or infinitely many triples such that c>rad(abc)1+ε, but why in the world would anyone want to?
I don’t have much of a background in number theory, so to write this post, I looked at a lot of examples to start wrapping my mind around the concept of the radical and the conjecture itself. I wanted to understand what rad(abc) was measuring and what an abc-hit meant qualitatively. As we’ve mentioned, two of the first hits are (1,8,9) and (5,27,32). In each case, we have two perfect powers (of different primes; 8=23 vs. 9=32 and 27=33 vs. 32=25) that are very close to each other.**
But abc-hits don’t have to be like that. We also have (1,48,49) and (1,4095,4096). In these cases, 48 and 4095 aren’t perfect powers of one prime number, but they do have fairly small prime factors and repeated factors: 48=24×3, and 4095=32×5×7×13, as we saw before. I came to the conclusion that, in an impressionistic way, we’re trying to find numbers that happen to be close together (that is, the smallest of the three numbers is often quite a bit smaller than the larger two) and are themselves or are divisible by large powers of primes.
After reflecting on some abc-hits, I started to understand how the abc conjecture could be related to other questions in number theory. For example, we can circle back to Fermat’s Last Theorem, the one about the equation an+bn=cn. It says that nth powers of a number never differ by nth powers. But could they be close to differing by an nth power, or could they differ by an mth power for some different number m? We could ask about similar equations: an-bm=1 or an+bn=5cn.** There are many, many ways of varying these and similar equations, collected under the umbrella term Diophantine equations, and the abc conjecture is a very general question that ends up addressing whether and how many of these Diophantine equations have integer solutions.
It’s unclear how long it will take for the mathematical world to fully understand and verify or find a flaw in Mochizuki’s claimed proof of the abc conjecture, but I hope that after this tour, the conjecture itself makes a little more sense, and you can start playing with some high-quality abc-hits for yourself if you want.
*This sentence has been updated for clarity. Thanks to the Twitter follower who pointed out the issue with the original sentence.
**These sentences have been corrected. Thanks to the Twitter follower who pointed out my errors.
Much has been written about the abc conjecture. If you’d like to read more about it, here are some links I recommend.
Bart de Smit has a page about abc-hits, including a table of the smallest 418 examples and a link to all of the triples composed of integers with under 18 digits.
For those with a strong math background, Andrew Granville wrote about how the abc conjecture is related to other important theorems in number theory in 2002.