Category Archives: number theory

A new tricubic sum for three!

Posted on September 24, 2019 by Brent

Here’s a nice Numberphile interview with Andrew Booker about the new discovery. They also talk about Hilbert’s tenth problem, undecidability, the reasons for doing computer searches like this, the role of science communication (such as Numberphile) in spurring discovery, and … Continue reading →

Posted in number theory | Tagged cubes, multiple, number theory, representation, sum | Comments Off

Sums of cubes: multiple representations

Posted on September 16, 2019 by Brent

I’m continuing a short series of posts on representing numbers as a sum of three cubes; previous posts are 33 is the sum of three cubes and More sums of three cubes. We now know that every number less than … Continue reading →

Posted in number theory | Tagged cubes, multiple, number theory, representation, sum | 1 Comment

More sums of three cubes

Posted on September 13, 2019 by Brent

About six months ago I wrote about the recent discovery that 33 can be written as the sum of three cubes. At that time, the only remaining number less than 100 whose status was still unknown was 42. And just … Continue reading →

Posted in number theory | Tagged cubes, number theory, sum | 6 Comments

Post without words #26

Posted on May 6, 2019 by Brent

Posted in modular arithmetic, number theory, posts without words | Tagged Euler, grid, totient | 2 Comments

Chinese Remainder Theorem proof

Posted on April 10, 2019 by Brent

In my previous post I stated the Chinese Remainder Theorem, which says that if and are relatively prime, then the function is a bijection between the set and the set of pairs (remember that the notation means the set ). … Continue reading →

Posted in modular arithmetic, number theory | Tagged Chinese, proof, remainder, theorem | 2 Comments

More words about PWW #25: The Chinese Remainder Theorem

Posted on April 5, 2019 by Brent

In a previous post I made images like this: And then in the next post I explained how I made the images: starting in the upper left corner of a grid, put consecutive numbers along a diagonal line, wrapping around … Continue reading →

Posted in modular arithmetic, number theory, posts without words | Tagged Chinese, grid, remainder, theorem, torus | 3 Comments

33 is the sum of three cubes

Posted on March 30, 2019 by Brent

I’m a bit late to the party, but I find this fascinating: we now know (thanks to a discovery of Andrew R. Booker) that the number 33 can be written as the sum of three cubes. This may sound unremarkable, … Continue reading →

Posted in number theory | Tagged cubes, number theory, sum | Comments Off

A few words about PWW #25

Posted on March 9, 2019 by Brent

In my previous post I made images like this: What’s going on? Well, first, it’s easy to notice that each grid starts with in the upper-left square; is one square down and to the right of , then is one … Continue reading →

Posted in modular arithmetic, number theory, posts without words | Tagged Chinese, grid, remainder, theorem, torus | 4 Comments

Post without words #25

Posted on March 5, 2019 by Brent

Posted in modular arithmetic, number theory, posts without words | Tagged grid | 4 Comments

Finding the repetend length of a decimal expansion

Posted on February 8, 2019 by Brent

We’re still trying to find the prefix length and repetend length of the decimal expansion of a fraction , that is, the length of the part before it starts repeating, and the length of the repeating part. In my previous … Continue reading →

Posted in computation, group theory, modular arithmetic, number theory, pattern | Tagged decimal, expansion, group theory, rational, repeating, repetend, totient | Comments Off