Another quick one today, but still interesting. Let’s start by noting that $latex 2018=6^4+5^4+3^4+2^4=13^2+43^2=44^2+9^2+1^2$. It’s the sum of four fourth powers, two squares, or three squares. How many numbers can there be like that? As it turns out, there are an infinite number of them. Let’s start by noting that there are an infinite number … More Squares and Squares and More Squares
We’ve all seen a magic square. It looks like this: 8 1 6 3 5 7 4 9 2 The numbers in each row, column, and diagonal add up to . What I want to do today is give you a formula for finding the magic constant for the normal magic square. What do those words mean? … More Magic!
Imagine you’re starting a country from scratch. You’re given a couple of trillion pounds, a few billion people, a large plot of land somewhere, and told to get on with it. In this admittedly unlikely scenario, what’s the first thing you do? Setting up a voting system is probably fairly high on the list. Today, … More Slings and Arrows
Here’s a question for you: what’s a decimal? When I write , what does the dot between the three and the one mean? For starters, it means I’m not French — the French use commas and dots the other way around, so they would write where we would write . Francophonic ramblings beyond, though, what’s … More The Point of No Return
For the last two weeks, we’ve been looking at perfect numbers and Mersenne primes. Today, I’d like to leave the perfect numbers aside for a minute and look at properties of Mersenne primes themselves for a minute. More generally, I’d like to examine numbers of the form and see which of them are prime. We … More Wait! Wait! Come back!
Welcome back. Last week, I introduced the idea of a perfect number, that is, a number whose divisors add up to twice itself. For example, the divisors of are , and . I also mentioned a Mersenne prime: a prime number of the form . Finally, I proved that if is a Mersenne prime, then is a … More The Most Perfect Blog Post: Part 2
Howdy folks! This one’s a fairly standard result, but I think it bears repeating here anyway. Define as the sum of the divisors of . This just means that you take all the numbers that divide into and add them up. Let’s calculate . is divisible by , so . There are already a couple … More The Most Perfect Blog Post: Part 1