Category: Math

Computing Legendre and Jacobi symbols

In a earlier post I introduce the Legendre symbol where a is a positive integer and p is prime. It is defined to be 0 if a is a multiple of p, 1 if a has a square root mod p, and -1 otherwise. The Jacobi symbol is a generalization of the Legendre symbol and uses the same notation. It […]

RSA implementation flaws

Implementation flaws in RSA encryption make it less secure in practice than in theory. RSA encryption depends on 5 numbers: Large primes p and q The modulus n = pq Encryption key e Decryption key d The numbers p, q, and d are kept secret, and the numbers e and n are made public. The encryption method relies on the assumption that in practice one cannot […]

Exploring the sum-product conjecture

Quanta Magazine posted an article yesterday about the sum-product problem of Paul Erdős and Endre Szemerédi. This problem starts with a finite set of real numbers A then considers the size of the sets A+A and A*A. That is, if we add every element of A to every other element of A, how many distinct sums are there? If we […]

Soviet license plates and Kolmogorov complexity

Physicist Lev Landau used to play a mental game with Soviet license plates . The plates had the form of two digits, a dash, two more digits, and some letters. Rules of the game His game was to apply high school math operators to the numbers on both side of the dash so that the […]

Soviet license plates and Kolmogorov complexity

Physicist Lev Landau used to play a mental game with Soviet license plates . The plates had the form of two digits, a dash, two more digits, and some letters. Rules of the game His game was to apply high school math operators to the numbers on both side of the dash so that the […]

Economics, power laws, and hacking

Increasing costs impact some players more than others. Those who know about power laws and know how to prioritize are impacted less than those who naively believe everything is equally important. This post will look at economics and power laws in the context of password cracking. Increasing the cost of verifying a password does not […]

Varsity versus junior varsity sports

Last night my wife and I watched our daughter’s junior varsity soccer game. Several statistical questions came to mind. Larger schools tend to have better sports teams. If the talent distributions of a large school and a small school are the same, the larger school will have a better team because its players are the […]

The science of waiting in line

There’s a branch of math that studies how people wait in line: queueing theory. It’s not just about people standing in line, but about any system with clients and servers. An introduction to queueing theory, about what you’d learn in one or two lectures, is very valuable for understanding how the world around you works. […]

A convergence problem going around Twitter

Ten days ago, Fermat’s library posted a tweet saying that it is unknown whether the sum converges or diverges, stirring up a lot of discussion. Sam Walters has been part of this discussion and pointed to a paper that says this is known as the Flint Hills series. My first thought was to replace the […]

Big O tilde notation

There’s a variation on Landau’s big-O notation  that’s starting to become more common, one that puts a tilde on top of the O. At first it looks like a typo, a stray diacritic mark. What does that mean? In short, That is, big O tilde notation ignores logarithmic factors. For example, the FFT algorithm computes […]

Unstructured data is an oxymoron

Strictly speaking, “unstructured data” is a contradiction in terms. Data must have structure to be comprehensible. By “unstructured data” people usually mean data with a non-tabular structure. Tabular data is data that comes in tables. Each row corresponds to a subject, and each column corresponds to a kind of measurement. This is the easiest data to […]

How fast can you multiply really big numbers?

How long does it take to multiply very large integers? Using the algorithm you learned in elementary school, it takes O(n²) operations to multiply two n digit numbers. But for large enough numbers it pays to carry out multiplication very differently, using FFTs. If you’re multiplying integers with tens of thousands of decimal digits, the […]

Asteroids named after mathematicians

This evening I stumbled on the fact that John von Neumann and Fibonacci both have asteroids named after them. Then I wondered how many more famous mathematicians have asteroids named after them. As it turns out, most of them. I wrote a little script to search for the 100 greatest mathematicians (according to James Allen’s […]

Exponential sums in 2019

I’ve made a small change in my exponential sum page. I’ll need to give a little background before explaining the change. First of all, you can read exactly what these exponential sums are here. These plots can be periodic in two senses. The first is simply repeating the same sequence of points. The second is […]

Goldbach’s conjecture, Lagrange’s theorem, and 2019

The previous post showed how to find all groups whose order is a product of two primes using 2019 as an example. Here are a couple more observations along the same line, illustrating the Odd Goldbach Conjecture and Lagrange’s four square theorem with 2019. Odd Goldbach Conjecture Goldbach’s Conjecture says that every even number greater […]

Groups of order 2019

How many groups have 2019 elements? What are these groups? 2019 is a semiprime, i.e. the product of two primes, 3 and 673. For every semiprime s, there are either one or two distinct groups of order s. As explained here, if s = pq with p > q, all groups of order s are isomorphic if q is not a factor of p-1. […]

Check sums and error detection

The previous post looked at Crockford’s base 32 encoding, a minor variation on the way math conventionally represents base 32 numbers, with concessions for human use. By not using the letter O, for example, it avoids confusion with the digit 0. Crockford recommends the following check sum procedure, a simple error detection code: The check […]

Base 32 and base 64 encoding

Math has a conventional way to represent numbers in bases larger than 10, and software development has a couple variations on this theme that are only incidentally mathematical. Math convention By convention, math books typically represent numbers in bases larger than 10 by using letters as new digit symbols following 9. For example, base 16 […]

New prime record: 51st Mersenne prime discovered

A new prime record was announced yesterday. The largest known prime is now Written in hexadecimal the newly discovered prime is For decades the largest known prime has been a Mersenne prime because there’s an efficient test for checking whether a Mersenne number is prime. I explain the test here. There are now 51 known […]

Kepler and the contraction mapping theorem

The contraction mapping theorem says that if a function moves points closer together, then there must be some point the function doesn’t move. We’ll make this statement more precise and give a historically important application. Definitions and theorem A function f on a metric space X is a contraction if there exists a constant q with […]