Oops, so much for posting once a week! My excuse is that I’ve been hard at work on my book. Well, nothing to do but get right back at it. I promise* I will be better** about posting regularly*** from now on.
* I had my fingers crossed when I typed that. Seriously.
** For low values of ‘better’.
*** Some restrictions may apply.
Anyway, I thought I would begin an intermittent series describing some currently open (unsolved) problems in mathematics. It’s pretty fascinating that in mathematics (unlike in many other academic disciplines) it’s not too hard to come up with questions that are very easy to understand, but incredibly difficult to answer! It just goes to show that sometimes, the simplest questions end up being the deepest.
Today, I’m going to talk about the twin prime conjecture. A prime number, as you may recall, is a positive integer greater than 1 which has no divisors other than 1 and itself. For example, 17 is prime, since there is no number less than 17 that evenly divides it, but 18 is not prime (it is composite) since it is divisible by, for example, 3. The first few prime numbers are thus 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37… Prime numbers serve as building blocks for all the other integers, since any integer can be written uniquely as a product of primes (this is called the integer’s factorization).
Let’s look at the sequence of distances from each prime to the next: 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, … See a pattern? I sure don’t. In fact, this is one of the most enigmatic things about the prime numbers: although there are theorems describing approximately how often primes occur, there is no neat pattern (that anyone knows of) which describes exactly when primes will occur. As you go through bigger and bigger integers, primes tend to get rarer… yet there is still no pattern. On average, the higher you go, the farther apart the primes will be — yet every now and then, primes still occur right next to each other, like 29 and 31. Pairs of primes which are only two away from each other are called twin primes. Larger examples of twin primes include 101 and 103, 281 and 283, and the whopping
a gargantuan pair of twin primes with 58711 digits each! This is actually the largest known pair of twin primes, discovered just this year, on January 15. So, here’s the question: are there infinitely many pairs of twin primes, just like there are infinitely many prime numbers? Or do the primes eventually get so few and far between that the twin primes stop? No one knows! The twin prime conjecture (a conjecture is like a mathematical guess) states that there are an infinite number of twin primes — most mathematicians believe that it’s true, but no one has been able to prove it.
A challenge to conclude: we could define “triplet primes” as a set of three consecutive odd numbers p, p+2, and p+4 which are all prime. This wouldn’t be a very useful definition, however, since 3, 5, 7 is the only set of triplet primes. Why?