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
3 thoughts on “Even perfect numbers”
I am also just reading it – read about 100 pages by now and have to say: WOW! What a wonderful book with hundreds of concise and very valuable articles about all major areas of math. One of the best math books I have ever got my hands on.
John … I’m not sure I’ve dealt with perfect numbers before (even or odd), though there is an ancient bell of some sort ringing in my head. I am always looking for little ‘twisty’ questions to entice my kids to wander of the familiar paths that math students tread. Perhaps … Given the Mersenne prime m=127 show m(m+1)/2 is perfect. [I would show them the nature of a Mersenne Prime as well as a definition and example of a Perfect number.] … I would hope for them to arrive at some factored form along the line of
127(128)/2=127(64)=127(1)(2)(2)(2)(2)(2)(2) and from this determine the factors to be 127, 1, 2, 2^2, 2^3, 2^4, 2^5, 2^6, 127×2, 127×4, 127×8, 127×16, 127×32 (leaving out 127×64) and their sum to be 127+1+ 2+2^2+2^3+2^4+2^5+2^6+127(1+2+4+8+16+32)=8128
mark (high school teacher in Ontario)
Mark: Maybe you could lead them through a series of examples with larger and larger Mersenne primes and get them to guess Euclid’s theorem.