# Underlying Pattern Governing The Prime Divisors

When we think of prime numbers, the first thing that we tend to associate them with is randomness. Prime numbers are scattered all over the number line and there is no fixed formula that can tell you when the next one is going to occur. This has been used heavily by mathematicians and cryptographers to develop security systems for internet, banking, communication, and so on. Coming to the topic at hand, we are going to talk about prime divisors of a given number. Prime divisors of a number are divisors of that number that happen to be prime numbers. Big surprise, right? Alright, what’s so interesting about them?   Continue reading

# The Interesting Case Of Russell’s Paradox

The picture on the left is actually a joke related the topic of discussion here. It’ll make sense soon! Russian Dolls, also known as Matryoshka Dolls, are wooden dolls that are placed inside one another. You can read more about them here. Back in the late 1800s, Set Theory was really picking up pace and mathematicians were getting really intrigued by this field. They were devotedly working towards formalizing this field of study. The problem was that there were many loose definitions floating around and there wasn’t any concrete work towards formalizing it. It all started when naive set theory was being used to discuss the foundation of mathematics. Instead of describing set theory with formal logic, people were describing it informally using words. So why should we care about that? What’s wrong with using words to describe set theory?   Continue reading

# What’s So Interesting About The Prime Counting Function?

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

# 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