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Open Season: Prime Numbers (part 2)

In this short series of articles, I’m writing about mathematical questions we don’t know the answer to – which haven’t yet been proven or disproven. This is the third article in the series, and across two parts will discuss various open conjectures relating to prime numbers. This follows on from Open Season: Prime numbers (part 1).

So, we have a pretty good handle on how prime numbers are defined, how many of them there are, and how to check whether a number is prime. But what don’t we know? It turns out, quite a lot.

The Twin Prime conjecture is about twin primes, which are pairs of prime numbers which differ by two (examples include 3 & 5, 101 & 103, and a whole bunch of others). The conjecture states that there are infinitely many such pairs. But it’s unproven – nobody has been able to construct a proof like Euler’s proof for all prime numbers, nor has anybody managed to prove the opposite, that there are finitely many.

However, that doesn’t mean there’s no hope at all. Earlier this year, a huge leap was made in the search for twin primes, in that a mathematician from the University of New Hampshire came up with a proof of a related fact: he proved there are infinitely many pairs of primes less than 70,000,000 apart. This may not sound that impressive, but as at least one person remarked on Twitter, 70,000,000 is a lot closer to 2 than it is to infinity. In fact, the bound was subsequently reduced, through further doing of maths, until within roughly two months of the original announcement, the gap was down to around 60,000. It’s not clear whether the same method will allow this gap to be reduced all the way down to 2, but if progress is made you can be sure we’ll report it here.

But our lack of complete knowledge doesn’t stop with twin primes; there’s a whole slew of different types of prime number for which we not only don’t know how they’re distributed, or how to predict when the next one occurs, but also we can’t even prove that there are or aren’t infinitely many of them. Primes which are palindromes, like 101; primes which are also Fibonacci numbers, like 13; and Sophie Germain primes, which are primes $p$ for which $2p+1$ is also prime, and one of the few things in mathematics named after a woman – all of these sets are of indeterminate infiniteness.

My favourite of these is Wieferich primes, which are primes $p$ for which $p^2$ divides $2^{(p-1)} -1$. There are, and I’m not kidding, precisely two known examples of such a number, which are 1093 and 3511 (brilliantly, sequence A001220 in OEIS). In fact, they were recently crowned victorious in the Aperiodical’s very own integer sequence competition. The best fact about them is that if the abc conjecture, a major open conjecture in number theory, turns out to be true, that implies that there are infinitely many prime numbers NOT of this form. While this doesn’t stop there being infinitely many Wieferich primes, it does mean that there’s no point on the number line above which all primes are Wieferich primes. So a major breakthrough in mathematics would still give us basically no information about this set of numbers.

And then, of course, there’s the big hitters. The Goldbach Conjecture, a huge open conjecture posed by Christian Goldbach in 1742, states that every even number can be written as the sum of two prime numbers. While it’s quite a fun game when you’re bored to pick an even number and try to find two primes which sum to it, there just aren’t enough long car journeys for it to be proved exhaustively, and nobody has either found a proof that it’s possible for every even number, nor found a counterexample that can’t be expressed that way. All the even numbers have been checked up to $2^{60}$, a mind-bogglingly large number, but of course that’s not nearly enough.

Again, there’s some hope – while the Goldbach Conjecture itself remains untouched, the Weak Goldbach Conjecture, which states that every odd integer greater than 5 can be written as the sum of three prime numbers, was proved in May this year. While it’s strictly a less strong statement, it’s evidence that this type of statement can be proved using modern mathematical techniques.

Possibly the most significant open result involving prime numbers is the Riemann Hypothesis. The hypothesis states that all zeroes of the Riemann zeta-function (a function taking a complex number) which aren’t negative even integers, lie on the complex line at $Re(z) = \frac{1}{2}$. Again, the first ten thousand billion or so have been checked, and they’re all on the line, but nobody has a proof. While some progress has been made using various approaches (my personal favourite is random matrix theory), we still don’t know if this pattern continues forever.

Sorry, what does this have to do with prime numbers, you say? Well, the usual statement of the zeta function is as a sum over all integers:
$$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$$
But it can also be stated in terms of prime numbers:
$$\zeta(s) = \prod_{p\textrm{ prime}} \frac{1}{1-p^{-s}}$$
This is a product over all primes, and while it’s not practically possible to calculate the value of the function using this form, it means that if we can know where the prime numbers are – the structure or pattern they make in the number line – we can find all the values of $s$ for which $\zeta(s) =0$, and hence solving the Riemann hypothesis means we’ve solved a large piece of the puzzle of prime numbers.

It also means that if you know where all the prime numbers are, you can probably solve the Riemann hypothesis, and claim your million dollars from the Clay Institute. And that’s what it’s all about, really.


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