Okay, here we are! Back at it again! I’m really sorry. But I had a bunch of deadlines converge on me and I just couldn’t keep up with the workload. A lot of you guys might not know this. But I also, write an idea journal for myself. It is a less curated version of my blog where I just randomly scribble about flight patterns in Pacific Gulls to what would happen in a universe with negative gravity. These are all my personal projects that I like to keep alive for the purposes of my mental sanity.
Some of these ideas make it to the blog. But as anyone who has done any real science will know. There is no such things as an abundance of answers in this world. Mostly, there is an abundance of questions. Some of which, I don’t even know how I would begin to answer. I like to further divide these into two more categories: Good questions and Bad questions.
A lot of people don’t think there can be bad questions. In my experience, this is not true. I have often found that the overwhelming majority of questions that are asked by me and to me are bad questions.
You would think that I would judge the people asking these. But I cannot. Since I am party to it as well. I think that any person who has tried to learn anything truly new to them has asked, at the very least, a few bad questions.
The famous example I like to give is: “how many kilometers is the temperature?”. This is an example that is (if not very rigorous) simple enough that most people who see it at least understand the existence of such an error.
So yes, there are bad questions asked all the time and my journal is full of these. There are some questions that are good. But of these, many are the of the kind that I don’t have the faintest idea of how to approach. And then there are others that I actually solve, only to discover that faceless horde of the internet has already solved it for me.
I was thinking about number theory (As one does sometimes): the twin prime conjecture. A conjecture is something that appears to be true but doesn’t currently have a formal and rigorous mathematical proof.
Twin primes are prime numbers whose difference is 2. So we have 3 and 5, 5 and 7, 11 and 13, 17 and 19 and so on. The conjecture is that there are infinitely many twin primes.
No one has the faintest idea how to prove this tantalizingly obvious concept. And this is one of the best things about number theory as a branch of math- It is extremely easy to understand some of these problems. Even if they are mind-crushingly difficult to solve.
So where do I fit into this story? Well, last year at about 1 pm in the night I was lying in bed and thinking about this problem. And I was struck by an observation: Every twin pair of twin primes (other than 3 and 5) has a multiple of 6 between them. 5 and 7, 11 and 13, 17 and 19. Could this be a pattern?
And I was off! Thinking about this, more and more. It was such an elegant and subtle thing. I had to wake up and look at a list of twin primes on the internet and lo and behold! It was true, every twin prime sandwiched a multiple of 6 in between! It seemed like the universe had only one recipe for the sandwich of a twin prime. And the filling was always a multiple of 6!!
Then, an even more general observation struck me! Every prime number (larger than 3) had a multiple of 6 either after it or before it.
But an observation is no good in math unless you can prove it. So I had to get out my notebook and pencil (Yes I still use a pencil as an adult! So what?) and was quickly trying to find a proof for this.
There are a few ways that I could prove this. But as the benevolent blogger that I am, I will use the simplest way here.
Every number greater than five (which is the first prime number after 3), let’s call it ‘k’, can be written as one of the six: k= {6n, 6n+1, 6n+2, 6n+3, 6n+4, 6n+5}. (where n is a natural number)
Now for a number to be prime it cannot give a natural number on being divided by another natural number. (except one and the number itself)
So what happens with the way we have defined numbers now
a) (6n) / 6 = n and 6 / 2 = 3
b) (6n+2) / 2 = 3n+1
c) (6n+3) / 3 = 2n+1
d) (6n+4) / 2 = 3n+2
As you can see, in each of the four cases we have at least one natural number that can divide the original number ‘k’ to give another natural number, which is not 0 or ‘k’ itself. So these four cases are never prime.
The only exceptions are 6n + 1 and 6n + 5.
These are numbers that are always adjacent to 6n or 6(n+1). Both of which are multiples of 6!!
So there you have it. Proof! Prime numbers greater than 3 are indeed adjacent to a multiple of 6.
But, I could never figure out how to use this result for the more general twin prime problem despite several hours of thinking.
And then, severely sleep deprived at this point, I decided to google this. (Some would have said this should be the first step.)
But then, after googling this a bit. I came to the disappointing realization. I found this site (From the University of Tennessee at Martin), which called this “Perhaps the most rediscovered result about primes numbers”.
Which means that generations of students saw this seductive problem and came to this precise conclusion independently of each other and were all disappointed to see hordes of thinkers before them, had already considered and abandoned this seductive intellectual dead-end.
Was I disappointed in seeing this? No doubt! It was already 3 pm on a weeknight. I had sacrificed my sleep. I had been called an amateur mathematician by some stranger who didn’t even know of my existence. And most importantly I felt the weight of my ignorance and narcissism rest on me for that brief moment.
But I would be lying if I didn’t admit that I also felt an excitement in this feeling! I had my ignorance laid bare to me and this meant I could learn something new.
The best feelings that one can often hope for in a career of science is a sense of humility when confronted with the vastness of knowledge, the even greater sense of humility when confronted with the vastness of ignorance. and our improbable capability to reduce some of the latter.
The unknown is scary! And rightly so. For our cave-dwelling ancestors, the unknown could have meant death. Arguably this continues to be true in the modern world. But what is remarkable to me is the fact that we can choose to expand our sphere of knowledge more and more. Sometimes, through the work of thinkers long gone and other time through painstaking thought and effort and math (yes you usually need math).
The simplest way to get a child to sleep in a dark room is a night lamp. That is what knowledge is to ignorance. It can’t light everything. But it is enough for one room.
But coming back, this is the inevitable fate of most of the good questions I have in my journal. So you might ask what, if anything, did I achieve in making this observation and by the tiny proof I presented here. The answer is: I solved something on my own.
A Rubik’s cube is a puzzle that was solved long before I was even born. And there are a lot of people who will look up the algorithm to solve it so that they can amaze people at parties. I never understood that. To me, that is like buying a trophy from a gift shop instead of winning the trophy for yourself.
It is not the trophy but how you get it that makes it joyful and satisfying.
Don’t mistake me, I do think there is value in reading. I do a lot of that for a living (but you might have guessed, I am not a mathematician!). Ultimately science is about questioning existing ideas and verifying repeatable experiments as it is about reading great thinkers and their ideas.
And that is something I don’t want to forget. I continue to write in my journal even when I don’t write my blog posts. Because these blog-posts need to have a beginning, a middle and an end.
I don’t need to restrict myself in my journal. Often my journal entries will end with the phrase “dead end!”, “does this mean anything?”. Or worse it will end with, “question made no sense, apparently!”. I don’t post these on my blog very often. Because this blog is about ideas that I have been able to develop to some satisfying degree or have neat mental ‘box’ to put them in. A half-developed idea that was abandoned by me, doesn’t make for an engaging read.
But I do find great joy in just being baffled by simple things that exist in direct defiance of my understanding. And in willing a small minority of these into being categorized and labeled in term of something more familiar.
To some extent that is why I have chosen my career in science. But regardless of what you do. I’d definitely recommend keeping an idea journal. It’s an idea I picked up while checking out a documentary on the life of Leonardo da Vinci. Who did this as well, as have a lot of people.
Often our journals go into the territory of dear diary nonsense. And there is perhaps a place for a different personal-journal for that purpose. But I do recommend an idea-journal if you don’t already keep one. It is remarkably fun!
A small update: This is definitely going to be a weekly blog moving forward. it’s quite hard to keep up with my old schedule for a hobby-blog. I might even not post for a while sometimes. Hope you guys understand! I’m still figuring this all out!
Many of my readers like to binge the blog. So in the meantime, maybe you can check out some of my earlier posts. Feel free to go back to the first post and scroll through them all (the calendar in the sidebar should help)! Here are some of the popular ones though:
Textiquette: We all text let’s do it better!
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