In the previous post we saw why primes are so important, and used Euclid’s proof to show that there are infinitely many primes. We further conjectured that not only there are infinitely many primes, they are also “nicely” distributed among the integers.
More precisely, if we fix some integer 
Trying to look for other methods, we mix our prime numbers with a bit of calculus, and turn to Euler for some help.
Continue readingThe prime numbers are one of those basic, yet mysterious, sets in mathematics, that while we know much about them, there are still many interesting open questions waiting to be answered, including probably the most well known conjecture in mathematics – the Riemann conjecture.
While this conjecture, and the attempts at its solution, use very advanced mathematics, even the most basic questions about the prime numbers already have many interesting results and lead to interesting tools, and in this post and the next two (here and here), we will try to understand some of these just by trying to “count” primes.
The idea for this post came from a question I saw in a math help forum about improper integrals. While this problem has a very simple solution using basic tools in integral calculus, I want to show a more geometric approach which can be generalized much further (and for the Hebrew speakers among you, I made a video explaining this as well here ).
Continue readingImagine going to a new amusement park for the first time. Once you get there, you go to the first ride that you see, and when you finish it, you randomly choose one of the roads leaving it and follow it until you reach another ride. You continue like this for a long time – every time you go on a ride, then choose a random road (might even go back on the same road that you came) and walk on it until you reach your next ride.
After doing this for a long enough time, on which ride would you go on the most? Go on the least? And how are these questions connected to eigenvectors and eigenvalues?
The radar is a detection system that was developed before and during World War II for military uses, though by today it has many other applications including, for example, astronomical and geological research. The name radar is an acronym for RAdio Detection And Ranging, and as its name says, it uses radio waves to detect targets. In this post we will try to understand one of the basic ideas behind the operation of the radar, which while it seems quite simple at first glance, when we start to try to understand the details, we encounter an interesting problem. In order to solve it we will need to use a mathematical theorem first formulated by the Chinese mathematician Sunzi from the 3rd century AD.
For an English video version of this post – https://youtu.be/OoWoxBhixDI
For a Hebrew video version of this post – https://youtu.be/30CEOB_j3Ag
Continue readingMathematical research is something that most people don’t really understand. They can imagine someone in a lab mixing chemicals or doing experiments with some scientific machinery, but mathematical research?
The goal of this post is to share with you a small part of this process – how we start with a simple problem, what happens when we complicate this problem, and how eventually we get what mathematicians call a research problem. For those interested, I made a video about this subject which you can see here and a Hebrew version here.
This problem’s origin is in the well known game of billiard. While in a general billiard game there are many balls, we will simplify it and have a square billiard table and a single ball. We then hit the ball and assume that it moves in a constant speed (so it doesn’t lose any momentum), and when it hit the edges of the table, it bounces back they we expect it to.
In this post we consider an interesting mathematical process which can be easily simulated by a computer, and generates interesting pictures. A video version of this post can be seen in here (for now in Hebrew).
Let 
The “limit” of this process produces a very interesting picture, as can be seen below.
This process is called a random walk, since at each step we choose one of the vertices at random and continue according to this choice.
Continue readingWhich text gives us more information – the full body of work of Shakespeare, or 884,647 random words written by 1000 monkeys?
To answer this, we talk with our good friend Alice, and ask her to send both of these texts to her close friend Bob. It should seem reasonable that the more information “dense” a text is, the harder it is to send it. Since the time of Shakespeare, we invented the internet over which we can only send messages over the simple 0\1 alphabet so Alice’s job is to translate the text to this 0\1 alphabet. If she manages to find a good translation with fewer 0\1 bits, the easier her job is. If the original text contained a lot of redundancy, namely many parts of the text don’t contain “new” information, then we expect Alice could shorten the message a lot. Conversely, the more information the message has, the harder it is to compress it.
So now, Alice needs to chose for each English letter 
We came to the point where we have already seen how lattices appear naturally in problems arising from number theory. In this post we construct a nice space which parametrize the set of all lattices of a certain dimension, with particular emphasis on 2-dimensional lattices.
In the post about number theory and lattices, we tried to determine when is the Euclidean distance in 
![\mathbb{Z}[i], \mathbb{Z}[\omega]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BZ%7D%5Bi%5D%2C+%5Cmathbb%7BZ%7D%5B%5Comega%5D&bg=ffffff&fg=333333&s=0&c=20201002)
Ofir David 