The Fundamental Theorem of Arithmetic

It is not uncommon to hear operations on numbers, even the computations carried out in modern computers, as “mere arithmetic.” But arithmetic is hardly simple or obvious when one gets down to the fundamentals and realizes the structure that must be present in our number system in order for it to work they way we intuitively think it should work. Thus, today we consider the Fundamental Theorem of Arithmetic.

Every positive integer (except the number 1) can be represented in exactly one way apart from rearrangement as a product of one or more primes.

Thus every integer has a canonical representation as a product of powers of primes:

where p1 < p2 < … < pk are primes and the αi are positive integers; 1 is represented by the empty product. As example 12 = 22 × 3, and 666 = 2 × 32 × 37. Fairly familiar stuff for anyone who paid attention during grade school and secondary school math classes. But the theorem itself is not self-evident, it is something that had to be proven true in order for our everyday arithmetic to hold. A good article on why the theorem is not obvious can be found here. It also has implications beyond the natural numbers. If we extend the canonical representation to allow both positive and negative values for ai, we get the set of positive rational numbers (fractions), for which the theorem still holds. It can also be generalized to more exotic constructions, such as Gaussian Integers. Gaussian integers, often denoted ℤ[i] are complex numbers a + bi where i is the square root of -1, and a and b are integers. It can be shown that the fundamental theorem of arithmetic holds for Gaussian integers, and that there is a definite set of “Gaussian primes” just as there are prime numbers. But while plotting the prime numbers and looking for visual patterns is an exercise in frustration, the rotational nature of complex numbers (which we have discussed in a previous article) causes the Gaussian primes to fall into a visually interesting radial pattern:

With all the important and sometimes confounding properties of primes, having ways to visualize them is always intriguing.

Any mathematical construct (specifically, a “domain”) that obeys the fundamental theorem of arithmetic is known as a Euclidean Domain. (Note that this has very little to do with Euclidean spaces or the other uses of the term in geometry.) We can observe many more Euclidean domains, such as generalizing Guassian integers to other roots of 1. If we use the cube roots of 1, for example (yes, 1 has three cube roots), we get the set of Eisenstein Integers: numbers of the form a + bω, where a and b are integers and:

Like Gaussian primes, Eisenstein primes have a distinctive radial pattern when viewed on the complex plane. Whereas Guassian primes divide into quadrants, Eisenstein primes form a hexagonal pattern.

Note that while the generalization works for square roots of -1, cube roots of 1, etc., it doesn’t necessarily work for all roots of 1. Some of those sets will not form Euclidean domains.

We can also look beyond numbers to other mathematical entities that form Euclidean domains. One such example can be found in knot theory, which we discussed in an article a few years ago. Knots can be expressed as unique combinations of prime knots:

From here we can consider the implications for music of the Euclidean domains, the accompanying Euclidean algorithm for computing greatest common divisors in any of these domains. But that will be left as an exercise for another day.