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

Alien maths – we’re counting on it

Are we alone in the universe? The possibility that we aren’t has preoccupied us as a species for much of recent history, and one way or another we need to know. The problem is, there is a lot of space, and only so fast you can move around in it, so popping over to our nearest neighbouring star for a quick look around is off the table. We simply don’t know how to communicate or travel faster than light. Nor have we picked up any signals which are identifiable as any sort of message from little green men.

Therefore, perhaps our best chance of making contact with an alien species is to announce ourselves to the universe. If we send out messages to promising-seeming parts of space in the hope that someone will be there to receive them, we might just get a response.

But supposing our signals reach alien ears (or freaky antenna things or whatever), what hope do we have of them being understood? Sure, we might make signals which are recognised as deliberate (and not mistaken for more literal ‘messages from the stars’), but how will they get anything across to aliens whose language is entirely unknown to us?

Scientists in the ‘70s were asking themselves these very questions, and the most promising approach they came up with to get around this problem was one which used maths. In fact, it used an ingenious trick dating back all the way to the Ancient Greeks. The fruit of their labour, broadcast in 1974, was called the Arecibo message.

So, what is it? First off, the Arecibo designers gave up on the hope of sending a written message the aliens could read. Better to stick with pictures – you have to assume aliens will be pretty low down on the reading tree. But this still leaves a conundrum.

When you’re sending a message to space, you have to send a binary signal – a series of ‘1’s and ‘0’s (aka bits) which you hope will start to mean something when it’s processed on the other end. This is precisely how sending pictures over the internet or between computers works too – your message is turned into bits, beamed to the other computer, and then turned back.

And herein lies the problem; the aliens receiving the binary signal won’t have any idea what they’re supposed to do with the bits or how to piece the message back together to make a picture again. You’ve posted them a Lego set but no instructions, and even though they’ve got the bricks there’s no way they’ll figure out whether it was supposed to be built into a race car or a yellow castle. After all, they might not even know what those are!

The way around this is to make the process for turning the message into a picture as simple as possible, so the aliens will be able to guess it. And the way you turn the bits into a picture really is very simple – just write them out in a 23×73 grid, and colour in any square with a ‘1’ in it. Below is what you get (with added colour-coding – see below for what the different parts mean).

aricebo

White, top: The numbers 1 to 10, written in binary

Purple, top: The atomic numbers for the elements in DNA

Green: The nucleotides of our DNA

Blue/white, mid: A representation of the double helix of DNA. The middle column also says how may nucleotides are in it.

Red: A representation of a human with the world’s pointiest head, with the average height of a man to the left, and the population to the right.

Yellow: A representation of the solar system and the sizes of the planets, with Earth highlighted

Purple, bottom: A curved parabolic mirror like the one used to send the message, with two purple beams of light being reflected onto the mirror’s focus, and the telescope’s diameter shown in blue at the bottom.

Image credit: Arne Nordmann 

But how, you might ask, are the aliens supposed to figure out the 23×73 dimensions of the grid? Here is where Ancient Greek maths comes to save us.

The Arecibo message is 1679 bits long. That sounds random, but it is anything but – 1679 is actually the product of two numbers, 23 and 73. Sound familiar? That’s the dimensions of the picture! It’s precisely the fact that 1679 equals 23 times 73 that lets you write out the 1679 bits in a 23×73 grid.

You might be wondering why we used such weird numbers for the sizing. Couldn’t we have chosen nicer, rounder numbers for the picture, like 50×100 say? No. If we did that, the aliens might make a mistake like writing out the bits in a 5×1000 grid or a 500×10 grid, and this would still work numbers-wise because 50×100 = 5×1000 = 500×10.

The key here is that unlike 50 and 100, 23 and 73 are prime numbers. Primes are numbers which can only be divided by one and themselves, like 3 and 5. And most importantly, any number can be split up into primes in a unique way – for instance, 15 is 3×5, and there is no other way to get 15 by multiplying together prime numbers. Likewise, there is no other way to get 1679 than as 23 times 73. So, it is impossible for the aliens to make a mistake when they have to draw out the grid. The Lego set you posted may have no instructions, but you were careful to include parts which can only go together the right way.

An Ancient Greek called Euclid knew this key fact, that numbers split uniquely into primes, over two thousand years ago. The Arecibo designers are banking on the aliens being at least as good with numbers as he was, to be able to decipher the message. Given these are aliens who are capable of picking up a radio signal from space, it seems like a pretty safe bet that they can manage better than an ancient society which believed women have fewer teeth than men because a . It’s a gamble, and it relies on assumptions that the maths we’re interested in is what all species will be interested in – but then what part of blindly shooting intergalactic friend requests into space in the hope someone we’d want to know finds them wasn’t going to be a gamble?

Joe Double

Funbers 17

Yet another applicant for the title of ‘world’s unluckiest number’, 17 spells ‘I am dead’ when rearranged in Italian. It’s also the ‘world’s most popular random number’ according to scientists at MIT and the number of ‘givens’ at the beginning of a Sudoku game that are required for there to be only one possible way to solve the puzzle correctly…

You can listen to all of the Funbers episodes from BBC Radio Cambridgeshire and BBC Radio Oxford here.

Funbers 2

Funbers continues with a return to the integers and the number 2. Good and evil, love and hate, light and dark, friends and enemies, we like things that come in pairs – even the great William Shakespeare was a fan! And let’s not forget it takes two to tango…

You can listen to all of the funbers episodes from BBC Radio Cambridgeshire and BBC Radio Oxford here.

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