It’s a Sunday morning. Again, I overslept in the afternoon, and I’m still awake at 2 am. And, I guess I’ll be sharing some easy math, as I am too tired to be tackling advanced problems and too energetic to fall asleep. So, here’s some low-quality (or not) content.
When I was younger, I knew very little math. However, I did find math to be an intriguing subject, so I often played around with random numbers to obtain some interesting results.
I had a poor memory at the time (and also now), so I didn’t like to read textbooks or learn about various theorems. As a result, I played around with basic arithmetic to derive some nice-looking “theorems”, many of which turned out to be well-known by the math community, while others turned out to be simply incorrect. Still, I enjoyed my time playing around with mathematics without the pressure of learning it for exams.
I do feel that some of them showcase simple and elegant examples of mathematics, so I’ll share two of these mathematical miniatures below.
The first one was something I “discovered” sometime around 5th grade, regarding the sum of odd numbers. I can’t remember how and why I wanted to add those numbers, but it should be one of those days where my devices were confiscated and I chose to spend my time playing with my calculator instead of doing homework.
It goes something like this:
1 = 1
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
The pattern should be quite obvious now, which is
and we only consider nonnegative integers. The proof of this identity was quite simple, and it goes by expressing both sides algebraically using the expression for the sum of an arithmetic sequence on the left:
Pretty nice and simple proof if you ask me. And, that’s it!
My second “invention” was, unfortunately, falsified very quickly. At the time, I was learning about the infinity of primes in our school’s mathematics extension course, and I came across Euclid’s proof by contradiction.
The proof first assumes that the primes are finite, i.e., assuming that all the primes are
Then, by definition, the number
can be written as the product of numbers from
However, since the smallest prime is 2 and
this cannot be possible. It follows that the number of primes cannot be finite. Naturally, I conjectured that
should be a prime for any consecutive
starting from 2. Again, I did not know how to immediately prove this, but I tested several numbers in an attempt to validate my conjecture:
I stopped at this point, quite confident that the result was correct. I even wrote a small Python program (my first time learning programming!) to verify that these numbers were all primes.
Unfortunately, I stopped a bit too early; the next number in this sequence, 30031, is not a prime. It was truly quite disappointing, but, how could we have found out without having to test for the primality of all these values?
Intuitively, the conjecture would hold for smaller sequences of consecutive primes, because the value
is quite small. Thus, there is unlikely to be a prime which occurs between
and
Since the numbers are coprime, we must have
On the other hand, as the elements of the sequence increase, the primes get bigger and bigger. This means there may be a higher likelihood of another prime occurring between these two values, hence a higher likelihood of
being a multiple of some other prime.
At the time I was writing this story, during this very instant, it inspired me to think about another problem: when is this expression prime, and when is it composite?
If you test some larger numbers and run them through a prime sieve, you’ll realize that most larger values of this expression are composite.
To rigorize this intuition (as I usually like to do), we propose the following conjecture:
There exists a sufficiently large natural number
such that for any natural number
the expression
is always composite.
Proving this is not easy! Some of the greatest mathematicians and number theorists have been investigating the distribution of prime numbers for centuries. When we approach such a conjecture, we also have no idea whether we will succeed.
I am too lazy to get out pen and paper to actually solve this conjecture right now, but I may do so (and provide updates to this problem) after waking up. In the meantime, please explore it for yourself!
Looking back on these miniatures now, they do seem quite silly, considering that some of my current peers were already winning national mathematics competitions at that age, and I was still fascinated by basic algebraic manipulations Gauss probably did in his mother’s womb.
Regardless, it feels interesting to recall the mathematics you explored when you were young, partially because you can re-imagine the sensation of being a pseudo-genius, but also because it is still fun! It’s interesting how I’ve never grown out of these things, regardless of their difficulty.
While it feels much more rewarding to resolve challenging problems, such tiny artifacts are simple, elegant expressions. They’re pleasing to do, even just to look at. I mean, if you do stare at those brief mathematical morsels for some time, you may be carried into a whole other world, one abstract and pleasant, intangible but rigorous. It lies in the heart of mathematics: a world of beauty and simplicity, things that deviate from reality without concealing the essence of nature, and what matters most for me: things you can think about, at any place or time, as long as you wish.
