This post originally ran 31 August 2009.
Everybody knows that a prime number is one which can only be evenly divided by itself and 1. Thus, for example, 13 is prime because no number except itself and 1 divides it. There has always been interest in primes, and lots of fun to be had with them (especially in cryptography, where your author got his start). More than that, primes are foundational to mathematics, as the Fundamental Theorem of Arithmetic attests: every natural number greater than 1 can be written as a unique product of prime numbers: every number is constructed from the raw material of primes. Isn’t that cool?
There are an infinite number of primes (here’s Euclid’s slick proof); further, and most curiously, primes appear to be salted “randomly” throughout the numbers. By this, I mean that there does not (yet?) appear to be any known list of premises from which we can deduce the primes. Another way of saying this is that there is no known formula for generating the primes.1 The implications of this are deep and vast.
The best all-around book on primes is Ribenboim’s, Book of Prime Number Records: beware, however, that this book requires at least basic familiarity with number theory.
Primes, if you like, are stingy with their divisors; but there are other numbers which are profligate. Antiprimes, which are usually known by the less euphonious title “highly composite numbers,” are the opposite of primes in the sense that they are numbers which have the largest number of divisors. Incidentally, theories about this class of numbers first came from Ramanujan.
The definition of antiprime is this: an (integer) number that has a larger number of divisors than any number less than itself. The first antiprime is 1: it has one divisor and no integers less than itself. 2 is the second antiprime: it has two divisors (itself and 1), which is more than the number 1 had. 3 is not antiprime because it has two divisors, tied with the two divisors of 2—to be antiprime it needed to beat the number of divisors of all numbers less than itself and it didn’t. This makes 4 an anitprime because it has 3 divisors; one more than 3 or 2 had. The first few antiprimes are: 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260 (it’s fun to check some of these by hand).
Several of these are familiar, are they not? For example, most of us know that the Earth spins once around each day. It would be useful to divide this period into increments that are easily manageable. 10 such chunks seem ideal: call these chunks hours. There is a lot to recommend 10 as an hour base. 10 can be split into two even chunklettes, one for day and one night, for example. And finger counting is a trick easy to master.2 But what if we wanted to divide the day up into quarters for, say, standing watch? Can’t do it evenly with 10, because 4 doesn’t divide it. In fact, 10’s measly three other divisors (1, 2, and 5) limit its ability to make differently (integer) sized chunks.
12 certainly works: it rates a healthy six divisors (1,2,3,4,6), so we have a lot of room to play. But 24 is even better with eight (1,2,3,4,6,8,12). As a bonus, five divisors of 24 are themselves antiprimes: this gives us lots of room to maneuver if we want to carve up the day into manageable, manipulatable, and malleable pieces. And those first five divisors are almost the first five integers, the naturally divisors to any block (one-half, one-third, one-fourth, and so on). The Egyptians, Babylonians and other peoples who originated the 24-hour day might have thought about the number 24’s special properties.
Now, once you have hours, you still have to divide up the time inside an hour. Small, but not too small, chunks seem to be optimal. Minutes, and inside them seconds, as everybody knows, are used. There are 60 of each of them and 60 is, of course, anitprime. 100 isn’t wonderful because it can’t be, for example, divided by three. The Babylonians, who gave us these divisions, liked 60 astrologically: the heavens are divided into 60 (arc minutes) times 6 equals 360 degrees.
Also interesting is that most antiprimes are next to primes: 2 itself is prime, so start with 4: both of its neighbors are prime; so are both of 6’s and 12’s. Only one neighbor of 24’s and 36’s. The problem starts with 120: both of its mates are composite (119 is divided by 7 and 17, and 121 is divided by 11).
I’m not aware of the common use of antiprimes larger than 60. Do you know of any?
1 There are plenty of formulas to identify prime numbers, and even algorithms to generate some primes, but none that gives us all primes.
212-finger counting is also easy. Ignore your thumb and look at your fingers. Count sections of the fingers separated by the joints and viola! 12 emerges. It is said that Sumerians counted like this.