The Infinite Hunt for Twin Primes
+ a tricky number theory problem!
We’ve all learned about prime numbers in grade school. A prime number is a number which is divisible by 1 and itself.
A basic question you can ask about primes is: how many primes are there? The answer is:
There are infinitely many primes.
This theorem quite ancient: it was proven in Euclid’s Elements (among other places).
Let’s add a twist …
You might think that we got off easy. But if you take this theorem and modify it slightly, it goes from being easy to being ridiculously hard. Let's look at the following statement:
There are infinitely many primes that are two apart. (For example, 11 and 13, or 29 and 31.)
This turns out to be a very famous unsolved problem called the Twin Prime Conjecture.
Why is the twin prime conjecture so much harder than showing that there are infinitely many primes? Because they are about fundamentally different things.
The infinitude of primes asks us the question: "how many primes are there?"
Whereas the twin prime conjecture asks the much more fine-tuned question: "how are the primes spaced out?" This is tricky to answer because the primes are famous for their unpredictable distribution.
What do we know about the conjecture?
Until recently, the twin prime conjecture remained hopelessly out of reach. Consider the following question:
Does the gap between successive primes go to infinity?
That is, as you go further out in the number line, does the space between each prime and the next one go to infinity? The twin prime conjecture says that this should not be the case. It predicts that there should be infinitely primes that are two apart, so the gap between successive primes cannot go to infinity. But until recently, even this weaker claim was beyond the reach of modern techniques!
This all changed in 2014, when Yitang Zhang proved the following groundbreaking result: there are infinitely many pairs of primes that differ by at most 70 million. In particular, the gap between successive primes does not go to infinity.
In 2015, James Maynard published a paper which reduced the bound from 70 million to a much lower bound of 600. That is, Maynard showed that there are infinitely many pairs of primes that differ by at most 600.
Using Maynard's ideas, the Polymath project further improved the bound to 246. The twin prime conjecture says that this bound can be reduced to 2 — but this is wide open.
Links to learn more
In this issue, I wanted to share some links about the twin prime conjecture:
Check out this wonderful chalkboard talk by Terence Tao about the gaps between primes. Tao explains the progress towards the twin prime conjecture that has taken place over the years:
I really enjoyed this interview of James Maynard with Hannah Fry. The first third of the video is a presentation by James Maynard about primes, and the second half is a more personal interview about Maynard’s problem solving process. It showcased the human aspect of problem solving in math, so I’d really recommend watching it:
Problem of the Week
As always, here is a challenge problem for this week:
Find an integer k such that the numbers
are the squares of the squares of three consecutive terms of an arithmetic series.
If you have a solution to this challenge problem, submit it here for a chance to be featured in the next issue of this newsletter.
Solution from last week
See here for the solution to last week’s problem.
Thanks for reading and happy learning! Until next time,
Adithya