Monthly Archives: September 2019

A combinatorial proof: counting bad functions

Posted on September 29, 2019 by Brent

In a previous post we derived the following expression: . We are trying to show that , in order to show that starting with a sequence of consecutive th powers and repeatedly taking successive differences will always result in . … Continue reading →

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A combinatorial proof: functions and matchings

Posted on September 26, 2019 by Brent

We’re trying to prove the following equality (see my previous post for a recap of the story so far): In particular we’re trying to show that the two sides of this equation correspond to two different ways to count the … Continue reading →

Posted in arithmetic, combinatorics, proof | Tagged , , , , , | 5 Comments

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 →

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Human Randomness

Posted on September 23, 2019 by Brent

Randomness is hard for humans It is notoriously difficult for humans to come up with truly random numbers. I don’t really have any data I can point to in particular, but there are quite a few well-known phenomena that show … Continue reading →

Posted in computation, people | Tagged , , , | 27 Comments

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 →

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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 , , | 6 Comments

A combinatorial proof: the story so far

Posted on September 6, 2019 by Brent

In my last post I reintroduced this seemingly odd phenomenon: Start with consecutive integers and raise them all to the th power. Then repeatedly take pairwise differences (i.e. subtract the first from the second, and the second from the third, … Continue reading →

Posted in arithmetic, combinatorics, proof | Tagged , , , | 1 Comment