When people ask me what number theory (my research area) is, we invariably end up talking about primes. When they ask me what I like about my field, I tell them about how I love to see patterns and make connections–something I like about all of mathematics–and about how number theory in particular has problems that are simple to understand, but lead to deep and complex theories. These themes are visible in the three representations of primes in the garden.
The First Two Scenes
Chipmunks discovering prime numbers
Beside the hill, a pair of chipmunks are sorting the integers into a prime pile and a composite pile. They are understanding what it means to be prime, maybe exploring primes for the first time. (Stay tuned for more on this scene in a future blog post!)
Clever squirrels and the Sieve of Eratosthanes
In the middle of the garden, some squirrels are taking a slightly more advanced approach. Instead of thinking about each number one by one, they’re using the Sieve of Eratosthanes to find all the primes at once. They have eliminated the multiples of two, three, and five already, and a bonus feature of their special translucent sieves is that they can see the factorizations of composite numbers, just based on the color–since the sieve corresponding to multiples of 2 is red and the sieve corresponding to multiples of 3 is yellow, the multiples of 6 are orange!
A Path Develops
These two vignettes were in the very early plans for our installation–in fact, a version of the chipmunk scene was in the very first design proposed by Ingrid and Dominique at the 2020 JMM. I was excited to join the Garden group in part because that’s where the primes were.
As we started to flesh out the vision for the garden in March, I asked if we could add another layer to the prime explorations–a garden path showing the Sieve of Eratosthanes for the Gaussian integers, also known as Z[i]. I was thrilled that everyone agreed!
The Gaussian Integers
The Gaussian integers are a complex-number analog of the regular integers, and it is the set of complex numbers that can be written as a+bi, where a and b are regular integers. Just like the more familiar integers, some Gaussian integers can be factored: for example, 3+i = (1+2i)(1-i), or 5 = (1+2i)(1-2i). Others, like 1+2i, 1-i, and 1-2i, can’t. In the garden path, each tile removes multiples of another Gaussian integer prime, so the composite Gaussian integers are eventually stripped away.
Classifying the primes in Z[i] leads to questions about which integers can be written as the sum of two squares and more! What patterns do you see in the primes of Z[i]?