Everybody has a strange pastime. Whether it’s something routine, like watching really old cartoons, or something more exoticlike intentionally calling every barista you interact with “Greg”—these quirky habits have the tendency of inexplicably making you a little bit happier than you were before (and potentially causing others to look at you like you escaped from an insane asylum). The different kinds of weird hobbies available to us has grown exponentially in the modern age, and you can basically be sure that for any kind of bizarre activity, you can find a poorly maintained hobbyist forum from the early 2000’s for it. (Haven’t had any luck finding a community of people who intentionally misname baristas, though.)

Some of the oddest pastimes take something easy and come up with a way to make it really hard for no apparent reason. For example, ironing your clothes is usually very boring, but you’ll probably get a bunch of YouTube views if you film yourself doing it while hanging off of a cliff. For the people who pursue these, a lot of the motivation comes from being the first or only person to succesfully do that strange specific action; you can rest assured that the guy who created the largest ball of paint by repeatedly coating a baseball won’t have his record taken from him anytime soon, even though I can make an even bigger paint ball by just pouring a bunch of paint into a spherical tank and letting it dry.

For this blog post however, I want to focus on two particular odd pastimes of this type that piqued my interest, the first of which I like to call constrained Super Mario 64. The gist of it is that a very dedicated group of gamers decided they would try and see whether or not they could beat specific levels in Super Mario 64 without ever performing important actions, like moving the joystick or pressing the A button. In fact, one particular guy has been trying for years to solve the “open problem” of figuring out how to beat the entire game without ever pressing the A button. It may be the case that doing so is impossible, but no one can say the guy hasn’t tried: just take a look at his channel in the hyperlink above and you’ll see all of his insanely complex attempts at beating levels without pressing the A button.

My personal favorite, and the video that made this guy Internet famous, is his successful attempt at beating this one specific level without pressing the A button. (Technically, he’s left it pressed since before entering the level, but we don’t need to be too pedantic.) Words can’t describe the amount of effort, dedication, and ingenuity he spent on doing this: you’ll have to see this work of art unfold for yourself. The video explaining his techniques below is about a half-hour long; you can find the much shorter uncommentated version here. If you do decide to watch it though, buckle up.

I was literally more excited watching the execution of this than I was when they found the Higgs boson (and I saw it live!). The fact this man was not immediately hired by NASA to coordinate rocket launches after the making of this video convinced me that there is no such thing as cosmic justice. If I could take any one person on an all-expenses-paid trip with me to the Bahamas, I would either take this guy or my favorite barista Greg.

In any case, I have nowhere near the skill or technical know-how to play Super Mario 64 like this, and every problem of this type in constrained SM64 that’s considered difficult has probably already been described; after all, there are only so many buttons you can’t press. As a result, if I want to get famous off of a weird pastime, I need to find some other strange activity which has undiscovered problems to solve, and that brings me to the second topic of this blog post: number theory.

Logo

Number theory is the study of numbers (great writing, Arnaldo), in particular the study of groups of numbers and facts about them. Some facts are easy to show, and some aren’tbut luckily for me, number theory has a massive amount of undiscovered problems! See, number theory is just like constrained Super Mario 64; it is extremely difficult, very interesting, mostly fun, and largely pointless (except for some key applications in cryptography*). The key difference, though, is that there’s only so much Super Mario 64; there are no limits to the amount of numbers and number groups.

*If you get all 120 stars without pressing the A button, you can find Yoshi on the castle roof and he’ll give you the private key to an offshore cryptocurrency wallet.

Perhaps the best thing about problems in number theory is that, as long as it’s not easy and it’s not impossible, I can basically claim some arbitrary unsolved problem is as important as any other famous problem because no problems are really “important” in any concrete objective sense. It’s like saying that beating a Super Mario 64 level without moving the joystick is more important than beating it without pressing the A button; one or the other might be easier, but they’re both pretty damn impressive, and doing either is ultimately pointless.

Easier said than done, you might think. Well, why don’t we actually take a crack at finding an “important” number theory problem? Let’s give it a shot by following these key steps:

  1. Find topics that are “hot” in number theory.
  2. Find an arbitrary specific problem involving these “hot” topics.
  3. Show this problem isn’t easy or equivalent to another known “important” problem.

We first need to look at what’s “hot” in the field of number theory, and perhaps the hottest topic in number theory is the study of what are called prime numbers. (It’s so hot that Wikipedia has an entire section on unsolved prime number theory problems!) These are numbers that can’t be divided by any number other than 1 or themselves without creating a bunch of decimal gunk. An example of a big number that’s prime is 89: try dividing it by any number other than 1 or 89 and you’ll always get a number with stuff past the decimal point. For clarity, the first few prime numbers are:

P_{i} = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, ...]

Another hot topic is the Fibonacci numbers; these are a bunch of numbers on a list defined so that the next number on the list is equal to the sum of the two last numbers. By defining both the first and second Fibonacci numbers as 1, the list of Fibonacci numbers begins as:

F_{i} = [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...]

Both prime numbers and Fibonacci numbers have been studied to death, and pop up very often in pop-math books and related media; I even vaguely recall seeing the Fibonacci numbers show up in The Da Vinci Code*. However, one thing that isn’t known (and is considered a well-known “important” problem to number theorists) is whether or not there are an infinite number of prime numbers that are also Fibonacci numbers. We can certainly spot a few prime numbers in the starting Fibonacci numbers I listed: 2, 3, 5, 13, 89, and 233.

*I always assume that science and math topics reach “peak pop-sci” when featured in a Dan Brown book. I send Mr. Brown emails every day about how cool low-Reynolds number fluid dynamics is, but he hasn’t taken the bait yet.

Anyways, figuring out the number of primes that are also Fibonacci numbers is a well-known problem; in order to come up with a new problem, we need to be a little bit more specific. Let’s think about the following list of related (and completely arbitrarily defined) numbers:

Start the list off by picking some prime number a. Pick the next number on the list by finding the a-th Fibonacci number. Then find the Fibonacci number corresponding to that number and put it on the list.

That’s it! This is just a list of specific Fibonacci numbers. To get a more intuitive sense of this list of numbers, I’ll call this list of numbers the “pointless sequence” T and start rattling off the first couple of numbers on the list if I pick, say, a = 7:

T_{1} = 7

T_{2} = F_{7} = 13

T_{3} = F_{13} = 233

T_{4} = F_{233} = 2211236406303914545699412969744873993387956988653

Jeez, that got out of hand really quickly! It seems like our arbitrary list is pulling in big numbers even at the start. But that’s great for number theorists; the bigger the numbers involved, the more difficult it is to deal with them, and the more challenging and “important” a problem is. One thing you may notice is that, if I pick 2 or 3 as my starting value, this sequence of numbers will just eventually start spouting out 1 forever. If I picked 5, it would just keep spouting out 5 forever, but if I pick any prime number bigger than that, I’ll start seeing the crazy blow-up we saw for a = 7.

Another thing you may notice is that those first three numbers on our list for a = 7 are prime! (The fourth one unfortunately isn’t.) We can then ask pointless questions about this list of numbers and hope we hit on a tough one, like if there exists an a other than 5 so that every number on this list would be prime. Because mathematicians don’t like problems in the forms of questions, we can guess that this isn’t true and reduce our problem to answering whether or not the following pointless conjecture is true; I’ll even put it into videogame format to make it pop a bit more.

Pointless Conjecture

Now that we have Step 1 and 2 out of the way, let’s proceed to Step 3 and check if this problem is easy or equivalent to another problem, particularly the “important” problem of whether or not there are an infinite number of Fibonacci primes. If there was a finite number of Fibonacci primes, then this sequence would have to eventually hit a non-prime and our problem would be solved. (Lucky for us it’s unsolved!) However, if the number of Fibonacci primes was infinite, it wouldn’t tell us anything about whether or not our list of specific numbers would eventually have a non-prime, which means our problem isn’t equivalent. Score!

So we know verifying the conjecture isn’t equivalent to this other problem, and we know that showing it’s true isn’t easy (because we’d solve this other nasty problem about infinite Fibonacci primes if we did). However, we need to figure out if showing that it’s false is easy, and that isn’t something we can check straightforwardly; so we’ll just have to drop it onto a math forum like stackexchange and see if anyone berates us for wasting their time on an easy problem.

Once we’ve completed all three steps, now we have to go through the hard process of actually trying to solve it; and for that, there’s no steps or rules other than staying dedicated, being creative, and enjoying it every day. In my case, I think it’ll probably be best if I just stick to calling baristas Greg.

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