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Category Archives: number theory
A new tricubic sum for three!
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 on A new tricubic sum for three!
Sums of cubes: multiple representations
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
More sums of three cubes
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
Post without words #26
Posted in modular arithmetic, number theory, posts without words
Tagged Euler, grid, totient
2 Comments
Chinese Remainder Theorem proof
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
More words about PWW #25: The Chinese Remainder Theorem
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
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 on 33 is the sum of three cubes
A few words about PWW #25
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 upperleft 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
Finding the repetend length of a decimal expansion
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 on Finding the repetend length of a decimal expansion