Mathematicians have obsessed over prime numbers for centuries, and will continue to do so for the foreseeable future. Prime numbers are so enigmatic and fascinating that mathematicians just can’t stop thinking about them! Prime counting function is probably one of the most famous problems in this domain. This function simply counts the number of prime numbers less than or equal to a given number. Pretty straightforward! But why is this of such great importance? What are we going to do with this information? Continue reading

# Category Archives: Mathematics

# What Is Bayesian Information Criterion?

Let’s say you have a bunch of datapoints and you want to come up with a nice model for them. We want this model to satisfy all the points in the best possible way. If we do this, then we will be able to use a mathematical formula to extract information about unknown points. At the same time, we should make sure that we don’t overfit our model to these datapoints. If we overfit our model, then it will tune itself too much to our datapoints and perform poorly on unknown data. So how we pick the best model? Where do we draw the line? Continue reading

# Why Are They Called “Elliptic” Curves?

Have you heard of elliptic curves before? They are used extensively in number theory and cryptography. The reason elliptic curve cryptography is gaining popularity is because it’s fundamentally much stronger than the RSA algorithm, the algorithm that we all love and adore. If you don’t know what elliptic curves are, just google it and see what they look like. You are reading this sentence without googling it, aren’t you? Okay I’m going to assume that you know what elliptic curves look like. Do they look anything like ellipses? No! So why are they called “elliptic” curves? Continue reading

# What Is A Holomorphic Function?

How do you feel when see the term “holomorphic function”? It just feels like we shouldn’t be looking further into it, right? I mean, it looks like an esoteric mathematical concept that should remain in advanced textbooks. Interestingly enough, holomorphic functions are very useful in real life. Holomorphic functions are ubiquitous in the field of complex analysis. Just to clarify, “complex analysis” doesn’t refer to an analysis that’s complex or difficult. Instead, it refers to analysis of functions of complex numbers. Alright, so let’s go ahead and see how something like this can possibly be useful in real life, shall we? Continue reading

# How Do We Know That There Are Infinitely Many Prime Numbers?

There is a very famous theorem which says that there are infinitely many prime numbers. For people who are new to this, a prime number is a number that doesn’t have any divisors except for 1 and itself. For example, 11 is a prime number because it doesn’t have any divisors apart from 1 and 11. On the other hand, 12 is not a prime number because it is divisible by 1, 2, 3, 4, 6, and 12. Now how do we know that there are infinitely many primes? As numbers get bigger, they tend to have more divisors. So may be at some point, all the numbers can possibly start being composite and they will have a lot of divisors, right? We can delve into a deep mathematical proof to prove this, but let’s take a different route. Let’s see if we can prove this with logic, shall we? Continue reading

# What Is Zeta Function Regularization?

There is a popular mathematical result which says that the sum of all natural numbers is -1/12. I have discussed it in detail here. This looks very unintuitive to a first time observer. In fact, most people would say that this is some kind of mathematical trickery. How can a bunch of positive numbers sum up to a negative fraction, right? Actually, there is a very real purpose to this whole thing of adding up all the natural numbers to get a negative fraction as the result. However, our general sense tells us that this shouldn’t be possible. The discussion in one of my previous blog posts was about the mathematics involved in this result. This discussion is more about the underlying fundamentals and where these results come from. So how do we explain this situation? Where is it used in real life? Continue reading

# What Is Relative Entropy?

In this blog post, we will be using a bit of background from my previous blog post. If you are familiar with the basics of entropy coding, you should be fine. If not, you may want to quickly read through my previous blog post. So coming to the topic at hand, let’s continue our discussion on entropy coding. Let’s say we have a stream of English alphabets coming in, and you want to store them in the best possible way by consuming the least amount of space. So you go ahead and build your nifty entropy coder to take care of all this. But what if you don’t have access to all the data? How do you know what alphabet appears most frequently if you can’t access the full data? The problem now is that you cannot know for sure if you have chosen the best possible representation. Since you cannot wait forever, you just wait for the first ‘n’ alphabets and build your entropy coder hoping that the rest of the data will adhere to this distribution. Do we end up suffering in terms of compression by doing this? How do we measure the loss in quality? Continue reading