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# Author Archives: Brent

## A combinatorial proof: counting bad functions

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

Posted in arithmetic, combinatorics, proof
Tagged consecutive, difference, function, integers, matching, powers
1 Comment

## A combinatorial proof: functions and matchings

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 consecutive, difference, function, integers, matching, powers
5 Comments

## 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

## Human Randomness

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

## 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

## A combinatorial proof: the story so far

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 consecutive, difference, integers, powers
1 Comment