Last week, the news broke that Shinichi Mochizuki, a Japanese mathematician, has claimed a proof of the abc conjecture, otherwise known as the Oesterlé-Masser conjecture, causing widespread excitement among the mathematical community. But what exactly is the conjecture, and what does its proof mean?
The abc conjecture was, until last week, seen as one of mathematics’ greatest unsolved problems, and sometimes seen as the “Holy Grail” of Number Theory: so many open problems in Number Theory follow as nearly a direct consequence of the abc conjecture or can be reduced to it. Like many Number Theory problems, it can be stated in (relatively) simple terms, and you don’t need to be a highly specialised and educated mathematical genius to understand what the problem is. However, its proof is far from simple, and is only understandable to a handful of mathematicians in the world, and at the moment most likely only to Mochizuki himself. It may take a considerable amount of time for the 500 page document to be fully verified and confirmed correct, but today I’m going to discuss the problem and some of its consequences.
There are several different statements of the conjecture, but I am using the one which I think is simplest to understand and explain.
The problem starts off very innocuously.
It then puts a couple of restrictions on a b and c.
a, b and c must all be integers (i.e. this is a Diophantine problem)
a and b are coprime (they share no common factors)
The conjecture then concerns the “radical” of abc, noted rad(abc). The radical of a number is the product of its unique prime factors, so for example, to find the radical of 36, we first find its prime factorisation (2×2×3×3), and then just take one 2 and one 3, to get 2×3=6. So 6 is the radical of 36.
As another example, the radical of 174 would just be 174, as its prime factorisation is 2×3×29 – there are no powers to remove, as all its prime factors are unique. In contrast, the radical of 348 would be 174, as its prime factorisation is 22×3×29, so we can remove one of the 2′s.
The conjecture then defines an “abc triple” as being a set of numbers a b and c such that rad(abc)<c. As an example, a=1, b=2, c=3 is not an abc triple, as rad(1×2×3)=rad(6)=3×2=6, which is greater than c (which is 3 in this case) so this is not a triple. However, if we take a=1, b=8, c=9, then we can check whether that is a triple, bearing in mind that b=23and c=33.
Since 6<c, and c is 9, then this is true and we have an abc triple.
Some other abc triples include a=5, b=27, c=32 and a=1, b=80, c=81, but I’ll leave you to test those if you want to. In fact, you can prove that there are infinite abc triples based on the fact that a=1, b=9n-1, c=9n is an abc triple for all positive integer values of n.
The conjecture then provides a measure of the “quality”, labelled q, of an abc triple. Quality is essentially a measure of how much lower rad(abc) is than q, essentially how remarkable it is, and is defined as follows.
So essentially q is the power that a radical of a triple must be raised to to get c – the lower the radical and the higher c is, the higher the quality will be.
This is where we get to the actual conjecture itself:
“There exists a number g such that q<g for any value of a, b and c.”
This is to say there is an upper bound in quality. This has been assumed to be true since the conjecture’s formulation in the mid 1980′s, that’s what made it a conjecture, but never until now has it been proven. There was one previous claim that the proof of this was trivial, made by 5 researchers from Indiana named Jackson, Jackson, Jackson, Jackson and Jackson contested that the conjecture was analogous to “do-re-mi,” and that it was “simple as 1 2 3,” but this claim was refuted when it was found that 1 2 3 was not in fact an abc triple.
All jesting aside, out of the known triples, the highest quality is ≈ 1.629911684, so it does seem intuitive that q cannot increase to infinity, but that does not make the significance of the proof any less great.
A mathematical proof isn’t about knowing what, it is about knowing why – this is the greatest difference between maths and observational sciences. In maths something is useless and disregarded unless it can be absolutely and verifiably proved not through a mass of evidence, but by clear, logical steps. This is partly because maths is more abstract than other sciences, so it is not necessarily the knowledge itself that is important, but the methodology, and what the results will imply. In this case, the proof will almost definitely lead to a greatly simplified proof of Fermat’s Last Theorem, solved by Andrew Wiles in 1994, and could also lead directly to proofs of Hall’s Original Conjecture, the Vojta Conjecture, the Szpiro Conjecture and many others. In some senses this is the “Higgs Boson” moment of number theory: so many open problems will now become so much more accessible and the field now has the potential to completely revolutionise itself.
Perhaps the most important consequence is that it provides a greater understanding of the prime numbers, and why they are so special. Prime numbers remain one of the greatest unsolved mysteries in all of mathematics, not just number theory, and this is a major step on the road to a fuller understanding of them. In recent years, this has become an especially important area of research, as much of the cryptography techniques that ensure secure connections via the internet and other network connections rely on techniques that involve the properties of prime numbers, such as the RSA Cryptography system, and any breakthrough that is made in this field could make significant changes, for better or for worse, to the security systems we use everyday.
In conclusion, this discovery, if proved to be correct, could well be the biggest advance that number theory has seen in the past 350 years since the time of Fermat. We will have to wait to see if this is verified, but it could cause an explosion of proofs from number theory. Watch this space.
I hope you’ve enjoyed this post! If you did then please check out our last two posts:
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We have now nearly reached the climax of our Physics of Sport series with the arrival of the Olympics and Paralympics, so have a look at the posts in the series here.