Math Musings

In the 1800s, mathematicians knew how to solve equations like these starting from the linear equation. However, what about equations of higher degree? Degrees 5,6,7 and beyond? A young teenager at the time in France, Ēvariste Galois answered this question. . And to do so he used a tool that he called a “group” . Around this time, Carl Friedrich Gauss was making sensational discoveries of his own. He showed a new technique called modular arithmetic which helped him solve many problems in number theory. As it turned out, Modular arithmetic shared many similarities to the groups used by Galois. The 1800s also saw a revolution in geometry. For more than 2000 years, Euclid dominated the scene with his book – “The Elements” but mathematicians began to realize there are other geometries beyond the one devised by the ancient Greeks. It didn’t take long before groups were found to be a useful tool in studying these new geom

Singular value decomposition (SVD) is one of the jewels of linear algebra. In modern times it has found applications in machine learning and Artificial intelligence. The most important feature of SVD is that of dimension reduction so that we are able to make predictions on very large data sets with a small subspace of variables.   Let’s try to understand how it works through a real-life example. Netflix has many subscribers and they have a collection of movies. When we watch a movie, we often see similar movies being recommended to us next time we go to Netflix and we wonder how did they figure out the kind the movies I like! The answer is SVD . As an example, suppose Netflix has 5 movies and multiple subscribers. Netflix gets the data for every user with a rating going from 0 to 5 with 0 implying the user did not watch a particular movie and 5 implying the user has watched it multiple times. A sample data set will look like this.  Each row in this matrix A will cor

First, we need to understand what we mean by a square free number. It essentially means that a whole number is not divisible by the square of a prime number. Let’s take the example of 30. From fundamental theorem of arithmetic, we can write it uniquely in terms of prime numbers as 30= 2 × 3 × 5 Since none of the prime numbers are repeated, 30 is square free. Now suppose we look at number 12. 12= 2 × 2 × 3 In this case the prime number 2 is repeated and hence 12 is not square free . The question we have posed in this blog is -what is the probability that a randomly chosen whole number is square free is not very precise. The reason being that we know there are an infinitely many primes and there is no associated probability distribution. So, for our discussion we will assume that we are talking about a finite set of whole numbers say a million and ask the question what proportion of those are square free. In a more general sense, we can take a big number x

The most famous L function is perhaps the Riemann Zeta function which has a Dirichlet series (sum of integers) and an Euler product of primes. Where is a complex number. Furthermore, Riemann zeta function is convergent for all x   > 1 but Riemann also showed the amazing symmetry it has about the vertical line x =1/2 which allows us to find values everywhere except the singularity at x =1. Riemann hypothesis claims that all non-trivial Zeros lie on the line of symmetry at x = ½. Mathematicians have been trying to prove that since the 19 th century as the L function encodes crucial information about the distribution of prime numbers. One way is to study different L functions since they all have a sum of integers, an Euler product of primes and the line of symmetry just like the Riemann Zeta function. The hope is that by studying several of these L functions some pattern will emerge which will help prove the Riemann Hypothesis. The l

Collatz conjecture is perhaps the most elementary open math problem which mathematicians still do not have a clue how to prove or find a counter example despite hundreds of papers written on the subject! In 1937, German mathematician Lothar Collatz made the following observation Start with a whole number. If it’s even, divide by 2. If it’s odd, multiply by 3 and add 1. Repeat. Eventually, you’ll always end up at 1. In closed form, this can be written as Here are some examples, 20 ,10,5,16,8,4,2,1 17 ,52,26,13,40,20,10,5,16,8,4,2,1  27, 82,41,124,62…………91,274…9232……….16,8,4,2,1 The number of steps it takes to reach 1 is called the final stopping time . For example, when we start with 27 it takes 111 steps to reach 1. To Date, numbers have been checked out to 87×2 60 and no counter-example has been found to disprove the conjecture. However, extreme caution is advised to accept Collatz conjecture to be true. History is replete with many such

Prime numbers are the most fascinating to mathematicians. Understandably so, as they are the basic atoms which uniquely constitute each number ( Fundamental Theorem of Arithmetic ) and also have very important applications in cryptography which is largely responsible for our internet security.   Over the centuries, we have learned a lot about them, yet there is an abundance of problems we still cannot prove. In this article, we will highlight a few of them. Goldbach’s conjecture Every even integer greater than 2 can be expressed as the sum of two prime numbers. E.g. 34=3+31 But is it always true? Legendre’s conjecture , proposed by Adrien-Marie Legendre , states that there is a prime number between n 2 and (n+1) 2 for every positive integer n. For example, between 4 and 9 which are square of 2 and 3, we have 5 and 7 as prime numbers.   The conjecture is one of Landau’s problems (1912) on prime numbers; as of 2020, the conjecture has neither been prove