Source: Quanta, Mar 2017
Unique prime factorization ensures that each number in a number system can be built up from prime numbers in exactly one way. In a number ring that includes a √-5 (in practice, mathematicians often employ number systems that use the square roots of negative numbers), duplicity creeps in: 6 is both 2 x 3 and also (1 + √-5) x (1 – √-5). All four of those factors are prime in the new number ring, giving 6 a dual existence that just won’t do when you’re trying to nail things down mathematically.
Unique prime factorization is a way of constructing a number system from fundamental building blocks. Without it, proofs can turn leaky. Mixing roots with the regular numbers failed as an attack on Fermat’s Last Theorem, but as often happens in math, the way in which it failed was provocative. It launched an area of inquiry unto itself called algebraic number theory.
Today mathematicians are actively engaged in the study of “class numbers” of number systems. In their crudest form, they’re a rating of how badly a number system fails the test of unique prime factorization, depending on which roots get mixed in: A number system that gets a “1” has unique prime factorization; a system that gets a “2” misses unique prime factorization by a little; a system that gets a “7” misses it by a lot more.
On their face, you’d expect class numbers to be randomly distributed — that class number 5 occurs with the same frequency as class number 6, or that half of all class numbers are even. That’s not the case, though, and current research in the subject aims to understand why. Today mathematicians are circling in on the structure that underlies class numbers and inching closer to establishing the truth about long-conjectured values. It’s an effort that has generated insights about how numbers behave that go far beyond a proof of any one problem.
Today, research on class numbers remains inspired by Gauss, but much of it takes place in a context established in the late 1970s by the mathematicians Henri Cohen, emeritus professor of mathematics at the University of Bordeaux, and Hendrik Lenstra, who recently retired from Leiden University in the Netherlands. Together they formulated the Cohen-Lenstra heuristics, which are a series of predictions about how frequently particular kinds of class numbers should appear. For example, the heuristics predict that 43 percent of class numbers are divisible by 3 in situations where you’re adjoining square roots of negative numbers.
By the time Cohen and Lenstra made their predictions, computers made it possible to rapidly calculate class numbers for billions of different number rings. As a result, there is good experimental evidence to support the Cohen-Lenstra heuristics. However, knowing something with confidence is entirely different from proving it.
“Probably in other sciences this is where you’d be done. However, in math that’s just the beginning. Now we want to know for sure,” said Melanie Wood, a mathematician at the University of Wisconsin-Madison.