July 25, 2023

There are a few online resources that we’ll be using in our discussion today.

• Sieve of Eratosthenes
• Ulam Spiral

How can we find primes? How could we check if a large number is prime? How are the primes distributed throughout \(\mathbb{Z}\)? These are a few of the questions we grappled with in our discussion today.

Quick facts. In no particular order, we recognized some important facts about the integers:

• There are infinitely many primes.
• For any \(n\in \mathbb{N}\), there exists some prime \(p\) so that \(p\mid n\). (Every number has a prime divisor)
• Fundamental Theorem of Arithmetic: All \(n\in\mathbb{N}\) can be written uniquely as \(n=p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}\), where \(p_i\) are primes and \(e_i\) are positive integers.

Unfortunately, none of these facts tell us about how to find primes.

In the spirit of prime-hunting, we talked about the Sieve of Eratosthenes (using this tool) to find all of the primes less than 100. We saw that only the multiples of \(2, 3, 5, 7\) had to be crossed out to obtain all of the primes from 0 to 100. This holds in general:

Fact. \(n > 1\) is prime if and only if for all primes \(p_i \leq \sqrt{n}\), \(p_i\) does not divide \(n\).

We also looked at the distribution of prime numbers in \(\mathbb{Z}\). From the Sieve of Eratosthenes, we saw that there are 25 primes between 0 and 100, and 21 between 100 and 200– so it seems like larger primes become more spread out. We used the Ulam spiral to attempt to visualize how the primes are distributed. The class made a few important observations:

• It looks more concentrated in the middle.
• As you go out to the edges of the picture, it looks less concentrated.
• It’s kind of a mess!
• Line-ish looking patterns appear.

Let \(\pi(n)\) be the number of primes less than or equal to \(n\). Perhaps by studying this, we can verify some of the observations above.

Theorem. (The Prime Number Theorem) \(\pi(n) \sim \frac{n}{\log n}\). (From our previous observations, what does “\(\sim\)” mean?)

Zeta function stuff will be put here soon.