I just got a review copy of Maths 1001 by Richard Elwes. As the title may suggest, the book is a collection 1001 little math articles. (Or “maths articles” as the author would say since he’s English.) Most of the articles are elementary though some are an introduction to advanced topics. Here’s something I learned from an article that was somewhere in the middle, the connection between perfect numbers and Mersenne primes.
Euclid (fl. 300 BC) proved that if M is a Mersenne prime then M(M+1)/2 is perfect. (A number is “perfect” if it equals the sum of its divisors less than itself. For example, 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14. A Mersenne prime is a prime of the form 2n – 1.) Euclid didn’t use the term “Mersenne prime” because Mersenne would come along nearly two millennia later, but that’s how we’d state Euclid’s result in modern terminology.
The converse of Euclid’s result is also true. If N is an even perfect number, then N = M(M+1)/2 where M is a Mersenne prime. Ibn Al-Haytham conjectured this result in the 10th century but it was first proved by Leonard Euler in the 18th century. (What about odd perfect numbers? See the next post.)
I’ve enjoyed reading Maths 1001. I’ll flip through a few pages thinking the material is all familiar but then something like the story above will stand out.
Update: Richard Elwes informs me that his book is published under the title Mathematics 1001 in the US. My review copy was a British edition.
Related: Applied number theory