Goldbach’s Conjecture: easy but hard

Often in Mathematics problems that are easy to state turn out to be extremely difficult to solve. Two hundred and seventy-five years ago, Goldbach wrote a letter to the famous Swiss mathematician Leonhard Euler in which he wrote the simple statement:

“Every even integer greater than 2 can be expressed as the sum of two primes.”

Just in case you are not up to speed with your maths (and let’s face it why would you be if you’re not a mathematician), let’s break this statement down. The even integers are the numbers divisible by two: 2, 4, 6, 8, …, 256, … and so on. The prime numbers are the ones that can only be obtained by multiplying one by themselves. For example, 3 and 5 are prime numbers because 3=1×3 and 5=1×5 and they have no other representations as a product of two numbers. However, 6 for instance is not prime because 6=1×6=2×3. In fact, all even integers, greater than 2 that were mentioned above, are not primes because they are all divisible by 2 and therefore can be represented as a product of two numbers in at least two ways: 4=1×4=2×2, 6=1×6=2×3, 8=1×8=2×4 etc.

And so, to Goldbach’s conjecture. It says that all even numbers: 4, 6, 8, 10, … can be written as a sum of two primes. Let’s see a couple of examples:

4=2+2

6=3+3

8=3+5

10=3+7

12=5+7

….

A nice way to represent the conjecture visually is through a “pyramid” and because we all love pretty pictures let’s see how this magic happens.

First, we write all of the prime numbers on two of the sides of a triangle as below: 2, 3, 5, 7 etc. We then draw a line leaving each prime number which is parallel to the opposite side of the triangle (stick with me), and finally at the points of intersection of these lines, we write the sum of the numbers. It sounds more complicated than it is as you’ll see with the following example. In the picture below, take the blue line coming out of the number 7 on the left and the red line coming out from the number 11 on the right. They intersect at 18 because 11+7=18. This means that the even integer 18 can be represented as a sum of the two prime numbers 11 and 7. If you look at the intersections of all of the red and blue lines in the pyramid, you’ll see that we actually get all of the even numbers. In other words, any even integer can be written as the sum of two prime numbers, and we can see what those two numbers are by finding the corresponding intersection on our diagram. This is Goldbach’s Conjecture.

goldbach

It is not very difficult to show that a small even number greater than 2 is the sum of two prime numbers – either by finding the corresponding point on the picture or by trying all of the possibilities. Let’s take 96. We start by checking the smallest prime number 3. 96=3+93, but 93 is not a prime, because 93=1×93=3×31. We continue with the next prime – 5. 96=5+91, which again doesn’t work because 91=1×91=7×13. Next, we try with 7: 96=7+89. Since 89 is a prime, we have obtained a representation of the number 96 as a sum of two primes.

We were able to quickly check whether 96 satisfies Goldbach’s conjecture because the number is relatively small. It becomes much harder to make these checks for larger numbers. It’s been verified with the use of a computer that the conjecture is true for numbers as big as 4×1018 and this is why the conjecture is believed to be true, but we do not yet have a formal mathematical proof. And being mathematicians, we cannot say something is true until we can prove it.

There have of course been many efforts over the last 275 years to try to prove the conjecture, most of which followed one of two routes. Either by proving that all even integers can be represented as a sum of some number of primes – as a sum of 6 primes (1995, Ramare) and as a sum of 4 primes (Herald Helfgott) – or by proving that almost all even integers can be written as a sum of 2 primes. But, as of yet, the secret formula required to unlock the proof of Goldbach’s Conjecture remains elusive.

You may be wondering why on earth mathematicians are spending their time and effort to prove this seemingly random result about prime numbers? Is it really that important? Whilst you may have a valid point about the applications of this particular conjecture, the value in proving such a result is not in the statement itself, but rather in the new methods, theories and techniques that will need to be developed to solve the problem. So, in 20, 10 or even 2 years from now when you hear that Goldbach’s conjecture has been proved, you should be happy not because we now know for sure that it’s true, but rather because some incredible new area of mathematics has been developed in the process. And who knows, this new area of maths may even pose a new, even more complicated conjecture that will occupy mathematicians for the next 275 years…

Mariya Delyakova

Maths proves that maths isn’t boring

If all the maths you’d ever seen was at school, then you’d be forgiven for thinking numbers were boring things that only a cold calculating robot could truly love. But, there is a mathematical proof that you’d be wrong: Gödel’s incompleteness theorem. It comes from a weird part of maths history which ended with a guy called Kurt Gödel proving that to do maths, you have to take thrill-seeking risks in a way a mindless robot never could, no matter how smart it was.

The weirdness begins with philosophers deciding to have a go at maths. Philosophers love (and envy) maths because they love certainty. No coincidence that Descartes, the guy you have to thank for x-y graphs, was also the genius who proved to himself that he actually existed and wasn’t just a dream (after all, who else would be the one worrying about being a dream?). Maths is great for worriers like him, because there’s no question of who is right and who is wrong – show a mathematician a watertight proof of your claim and they’ll stop arguing with you and go away (disclaimer: this may not to work with maths teachers…).

However, being philosophers, they eventually found a reason to worry again. After all, maths proofs can’t just start from nothing, they need some assumptions. If these are wrong, then the proof is no good. Most of the time, the assumptions will have proofs of their own, but as anyone who has argued with a child will know, eventually the buck has to stop somewhere. (“Why can’t I play Mario?” “Because it’s your bedtime.” “Why is it bedtime?!” “BECAUSE I SAY SO!”) Otherwise, you go on giving explanations forever.

The way this usually works for maths, is mathematicians agree on some excruciatingly obvious facts to leave unproved, called axioms. Think “1+1=2”, but then even more obvious than that (in fact, Bertrand Russell spent hundreds of pages proving that 1+1=2 from these stupidly basic facts!). This means that mathematicians can go about happily proving stuff from their axioms, and stop worrying. Peak boring maths.

But the philosophers still weren’t happy. Mostly, it was because the mathematicians massively screwed up their first go at thinking of obvious ‘facts’. How massively? The ‘facts’ they chose turned out to be nonsense. We know this because they told us things which flat-out contradicted each other. You could use them to ‘prove’ anything you like – and the opposite at the same time. You could ‘prove’ that God exists, and that He doesn’t – and no matter which one of those you think is true, we can all agree that they can’t both be right! In other words, the axioms the mathematicians chose were inconsistent.

Philosophers’ trust in maths was shattered (after all, it was their job to prove ridiculous stuff). Before they could trust another axiom ever again, they wanted some cast-iron proof that they weren’t going to be taken for another ride by the new axioms. But where could this proof start off? If we had to come up with a whole other list of axioms for it, then we’d need a proof for them too… This was all a bit of a headache.

The only way out the mathematicians and philosophers could see was to look for a proof that the new axioms were consistent, using only those new axioms themselves. This turned out to be very, very hard. In fact (and this is where Gödel steps in) it turned out to be impossible.

Cue Gödel’s incompleteness theorem. It says that any axioms that you can think of are either inconsistent – nonsense – or aren’t good enough to answer all of your maths questions. And, sadly, one of those questions has to be whether the axioms are inconsistent. In short, all good axioms are incomplete.

This may sound bad, but it’s really an exciting thing. It means that if you want to do maths, you really do have to take big risks, and be prepared to see your whole house of cards fall down in one big inconsistent pile of nonsense at any time. That takes serious nerve. It also means mathematicians have the best job security on the planet. If you could just write down axioms and get proof after proof out of them, like a production line, then you could easily make a mindless robot or a glorified calculator sit down and do it. But thanks to Gödel’s incompleteness theorem, we know for sure that will never happen. Maths needs a creative touch – a willingness to stick your neck out and try new axioms just to see what will happen – that no robot we can build will ever have.

Joe Double

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