Gems in STEM: A Discussion of Primes
Hi everyone! I’m Apoorva, a high-schooler who enjoys mathematics (and terrible jokes, as seen from the title). I’ve loved math ever since I was a kid, and have been involved in math competitions since elementary school. Additionally, I’ve co-authored two published papers in number theory. I’ll be using this platform to share STEM topics I’m excited about, and that I hope will excite you too. These articles are published in my local newspaper in my column, Gems in STEM, and I figured I might as well upload them here for anyone interested! Everything written is meant to be fairly accessible with the occasional advanced details weaved in. Future articles may include social issues in STEM, its intersections with other subjects, and various other topics that are prominent in these fields. I hope you enjoy today’s topic! (Note: Medium doesn’t have great support for equations and such, so some of the formatting may not be ideal.)
Today, we’re going to discuss one of the most popular areas of math: primes! To recap: a prime number is an integer (AKA a whole number like …,-2, -1, 0, 1, 2,…) greater than one with exactly two divisors: 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, and so on. Two is often jokingly called “the oddest prime number,” as it’s usually an outlier in theorems involving primes.
Okay, so? Why do we care about primes? We care because we can break down all numbers into their prime parts (known as the Fundamental Theorem of Arithmetic). For example, 60 = 2 ⋅ 2 ⋅ 3 ⋅ 5. This is called a prime factorization, and is actually unique! Can you convince yourself why?
Now, a natural question arises: how many primes are there? The answer is infinitely many! How do we know this for sure? Well, in order to convince everyone that their conjectures (prospective theorems) are not crazy, mathematicians write proofs. So, let’s prove this theorem! First, let’s assume we’re wrong and that there is a fixed, or finite, number of primes. Let’s say there are exactly k primes. We’ll put these primes in a set (you can think of it as a non-ordered list):
If we can somehow generate a number that cannot be factored by the primes in this list, we have proved this is an incomplete list of primes, which would show that there is an infinite number of primes. (Think about why this is true.) Continuing on, let’s construct the number
We know that n must have a prime factorization with at least one prime factor. However, n has a remainder of 1 when divided by any of the primes in our list, meaning none of the prime factors of n lives in our current list. So, we cannot construct a finite list of primes. Therefore, there must be infinitely many primes!
Now, things may start to get a little more technical, but I’ll try my best to frame it as simply as I can. The next question we can ask is, “How many primes are there that are less than or equal to a number x?” This is a significantly harder question, but still has an answer! First, define the prime counting function π(x) as the number of primes less than or equal to x. For example, π(13)=6, because we have six primes (2, 3, 5, 7, 11, 13) that are less than or equal to 13. Now, back to our question. The answer to this is known as the Prime Number Theorem, which states that π(x) is asymptotic to x/㏑(x), written
Here ㏑(x) denotes the natural logarithm of x. (Typically, mathematicians denote the natural logarithm as ㏒(x), but for some reason this isn’t the notation used in most schools.) For readers who know some calculus, this asymptotic means that, given functions f(x) and g(x), f(x) ~ g(x) is equivalent to
To understand this better, let’s think about it in terms of probability! Suppose we’re choosing a random number from the first x numbers. This theorem says that the probability of us choosing a prime number is roughly
or that a number x is prime “with probability” 1/㏑(x).
Now I want to leave you with a few questions to think about. We know that there are infinitely many primes, but what if we consider different subsets of primes? Are there infinitely many primes of the form 2^n- 1 for integers n>1? Or the form n^2+1? These simple-to-state questions about prime distribution are actually currently unsolved. Another popular question is the twin prime conjecture, which states that there are infinitely many pairs of primes that differ by 2. These unsuspecting questions reveal a hard truth: we really don’t know a lot about primes! If you’re currently thinking, “Well these don’t seem that hard,” prove one (and if you do manage to do this, mention me when you receive the Fields Medal)!
Finally, are you in your prime? Check your age to see! I’m 15, so unfortunately I’m not in my prime (as 15 is not prime), but I am in my semi-prime!
If you found this interesting, make sure to check out the next article! If you have any questions or comments, please email me at firstname.lastname@example.org. Until next time!