Failure Of Unique Factorization
A simple example of the failure of the fundamental theorem of arithmetic
Ernst Kummer was a German mathematician active in the early 1800s. He is most famous for his beautiful work on an approach designed to solve Fermat’s Last Theorem (FLT).
Today we will talk about a barrier that stopped his approach from succeeding.
I am currently teaching basic discrete mathematics at Tech. The first topic is elementary number theory, where we have covered some of the key properties of primes. Of course, we have covered the Fundamental Theorem of Arithmetic (FTA). Recall it says:
Theorem: Every natural number greater than is the product of primes, and moreover this decomposition is unique up to ordering of the primes.
This is a classic theorem, a theorem that is fundamental to almost all of number theory. It was implicit in Euclid’s famous work, and was perhaps stated and proved precisely first by Carl Gauss in his famous Disquisitiones Arithmeticae.
Alas as Kummer discovered the FTA fails for other sets of numbers. For other types of numbers half of the theorem still holds: every number that is not a generalization of can still be written as a product of “primes.” But uniqueness is no longer true.
In 1847, Gabriel Lamé claimed, in a talk to the Paris Academy, that he had solved FLT by using complex numbers—in particular numbers that were generated by the th roots of unity. Recall those are complex numbers that are solutions to the equation
Clearly is a solution always, but for primes there are other solutions: in total. Lamé’s argument was perfect, rather easy, used standard arguments, and was incorrect. The great mathematican Joseph Liouville at the talk questioned whether Lame’s assumption about unique factorization was justified; and if not, it followed that the proof was not valid. See here for a fuller discussion of this.
Liouville’s intuition was right. The FTA failed for Lamé’s numbers. Weeks after Lamé’s talk, it was discovered that Kummer had three years earlier shown that FTA indeed held for some primes but failed for . One might guess that the reason Lamé, and others, thought that factorization was unique for these numbers is that the first counterexample was for . Kummer famously showed that FTA could be replaced by a weaker and very useful statement, which allowed his methods to prove FLT in many cases. But not all.
Failure In Simpler Systems
The failure of FTA for what is now called cyclotomic integers is a well studied and important part of number theory. It is well beyond my introductory class in discrete mathematics. This failure has led to the discovery of related failures of uniqueness. One of the classic examples is the Hilbert Numbers—named after David Hilbert—of course.
Hilbert numbers are the set of natural numbers of the form . Thus they start:
Every Hilbert number greater than is the product of “Hilbert primes.” But note that a number can be prime now without being a real prime, that is a prime over all the natural numbers. Note that is a Hilbert prime: it cannot be factored as , since is not a Hilbert number.
The key observation is that some Hilbert numbers can be factored in more than one way:
Thus FTA fails for Hilbert numbers.
Another popular example of the failure of FTA uses square roots.
Note the numbers then are all those of the form:
where are integers. Many examples can be created in this way by changing to other integers. It is a major industry trying to understand which yield a set of numbers so that
satisfy the FTA.
Let us consider an extremely complex subset of the natural numbers:
Okay is just the set of all even numbers. This set is closed under addition and multiplication, and we claim that it has the following nice properties:
- The primes in are easy to describe;
- Every number in is a product of primes;
- The factorization in is not unique.
Let’s look at each of these in turn. A prime in is where is an odd natural number. The usual argument shows that every number in is a product of primes. For example:
Note, this process now stops, since is a prime. The last part showing that FTA fails in this setting is easy: Let be distinct odd primes. Then
Note, the factorizations are different. For example,
Does the simple example of even numbers help in understanding why FTA is special? Where was it first stated in the literature that even numbers fail to satisfy unique factorization? We cannot find it after a quick search of the web: all examples we can find are either the Hilbert numbers, some examples using square roots, or something even more complex.
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