2, 3, 5, 7, 11, … . What do these numbers have in common? They are all primes: they are not divisible by any number except 1 and themselves. That is an intriguing property for a number to have. It reminds us of the fundamental particles in physics; indivisible units that make up all the atoms and molecules we find in nature.
The prime numbers are the atoms of number theory.
And indeed, the primes do play such a fundamental role in mathematics. If you take a whole number that is not prime, say 12, you can break it down into its divisors, for example 12=2x6=3x4. Each divisor that is not prime itself can be further broken down: 12 = 2x6 = 2x2x3 and 12=3x4=3x2x2. Every divisor in this example is now prime, so we stop. For a number other than 12 you may have more steps to go, but ultimately you will also end up with an expression of your number as a product of primes. And what is more, this expression is unique: the primes might appear in a different order but any such expression of a number will always contain exactly the same primes.
This result, that every whole number is a product of primes in a unique way, is called the fundamental theorem of arithmetic. It made its earliest appearance in Euclid’s famous book The Elements in around 300BC. Euclid also showed that there are infinitely many primes: they never run out as you move up the number line.
So is that a done and dusted piece of mathematics then? Perhaps one might want to add to Euclid’s findings some recipe for spotting primes and, if a number isn’t a prime, a quick and easy way of factoring it into its prime components. But it turns out these simple requests land you right in the thicket of modern mathematics and computer science.
The problem is that there isn’t a simple recipe for spotting primes. Checking whether a given number is prime requires more and more calculational power as the number gets larger. Computer geeks have turned the quest for larger and larger primes into a mission. At the forefront is the GIMPS project, the longest-running distributed computing project in the world, which uses the donated power of thousands of idle computers to hunt for larger and larger primes. The record at the time of writing, discovered on January 25 2013, is a 17,425,170 digit number, more conveniently written as 257885161-1.
This may sound like frivolous fun (if you are that way inclined) but there is a serious aspect to such prime problems. While checking whether a number is prime is hard, factoring a non-prime number into its prime components is even harder when the number is large. The mechanisms that encrypt your bank details when you send them over the internet rely on this fact. The receiver takes two large primes and multiplies them, which is easy to do. It then sends the resulting number N to the sender, openly and publicly. The sender uses N to encrypt the message (using a standard method). Decrypting the message, however, requires knowledge of the prime factors of N. The receiver has that knowledge, that’s how it got N in the first place, but anyone intercepting the transfer would need an unfeasible amount of computing power (if the original primes are large enough) to factorise N and crack the code. So the safety of your bank details, and many other secret messages, depends on the inexhaustible supply of primes.
Back in the land of pure mathematics the primes are a hot topic for another reason: they lie at the heart of unanswered questions that have been bugging mathematicians for centuries. Some of those questions are so straightforward you might come up with them yourself with a bit of tinkering. For example, playing around with even numbers you might find that, as long as they are greater than two, they seem to always be the sum of two primes: 4=2+2, 6=3+3, 8=3+5, 10=3+7=5+5, and so on. This is true for any even number mathematicians have ever checked, so it is probably true for all of them. It would make a pleasing result, showing the primes are fundamental building blocks not just in a multiplicative, but also in an additive sense. The trouble is that so far nobody has been able to prove that this conjecture, so easily stated, is actually true. It has been attributed to the German mathematician Christian Goldbach, who first mentioned it in a letter in 1740, and it now carries his name: the Goldbach conjecture.
The even integers from 4 to 28 as sums of two primes.
Goldbach’s conjecture states that every even integer greater than 2 can be
expressed as the sum of two primes.
Do primes come in twins infinitely often?
Another thing you might notice is that primes often seem to come in pairs: 3 and 5, 5 and 7, 11 and 13, 17 and 19, and so on. Are there infinitely many such pairs? Can you always be sure to find another pair as you move up the number line? Again, the answer to this twin prime conjecture is unknown, despite the conjecture having been around for over 150 years.
The twin prime conjecture hints at a more general question: how are the primes sprinkled among the other numbers? In general they do seem to get sparser as you move up the number line and we know that, as well as twins, you can find arbitrarily long gaps between them. The prime number theorem gives an estimate of the number of primes there are below a given number n. It’s approximately n/ln(n), where ln(n) is the natural logarithm of n. This estimate becomes more and more accurate as n gets larger.
But can we say more? The answer lies concealed in what is perhaps the hardest open problem in mathematics: the Riemann hypothesis, named after the 19th century mathematician Bernhard Riemann.
Bernhard Riemann (1826-1866) formulated
the Riemann hypothesis one of the hardest open problems in mathematics.
The mathematics is complex, but a proof of the Riemann hypothesis would reveal much about the ebb and flow of primes on the number line. And it would win the person who finds it $1 million from the Clay Mathematics Institute, who offered one of its Millennium Prizes for its resolution. Those atoms of number theory, first discussed over 2000 years ago, still have a lot to offer today!
How many kisses do you get if every person in a room full of ten people kisses every other person once? The first person kisses 9 others, the second one kisses 8 others (having already kissed the first), the third kisses 7 others, and so on, with the 10th person not having any new person to kiss at all. So in total you get
9+8+7+ …. +2+1 = 45
kisses. You can also look at this another way. Every person kisses 9 others and there are 10 people, giving 10x9=90 kisses. But this counts every kiss twice, so we have to divide by 2 to get a total of
This is neat: imagine you have n+1 people, then using the two ways of counting tells us that
n+(n-1)+(n-2)+ … +2+1=(n+1)n/2
In other words, the kissing argument gives us a quick formula for summing the first n whole numbers: no need to tap them all into your calculator.
If you prefer your maths visually, there is another way of keeping tally of the kisses: ask everyone to make a cross on a piece of paper for each, as yet uncounted, kiss and make sure they arrange them as in the picture below:
Once all the kissing has been done the shape we see is… a triangle! It doesn’t matter what the value of n is, we will always get a triangle since we’ve been stacking kisses like oranges in a grocery shop. This is why numbers of the form (n+1)n/2, numbers that are the sum of the first n whole numbers, are called triangular numbers. Substituting n=1, n=2, n=3, and so on gives the sequence
1, 3, 6, 10, 15, 21, 28…
which adorns one of the faces of your Cube. The nth triangular number gives you the number of kisses between n+1 people.
If we go up a dimension, the kiss-stacking problem leads us straight to some cutting edge mathematics. Let’s swap kisses for oranges and ask: what is the most efficient way of stacking the oranges? Efficient in the sense that it leaves the smallest gaps possible? The question was posed in the 16th century by Sir Walter Raleigh (who also brought Europe the potato) to his assistant Thomas Harriot, although Raleigh was thinking of canon balls rather than oranges. Harriot eventually forwarded the question to Johannes Kepler, best known for his laws of planetary motion. Kepler suggested that one most efficient way of stacking the balls is the one commonly used by greengrocers. Create the first layer of oranges in a similar way as we created our kiss-triangle, with oranges in one row snuggling into the dip between two oranges in the adjacent row:
Then put the oranges in the next layer into the dip between three oranges in the layer below. Using this method you can fill just over 74% of space with oranges, canon balls, or whatever balls you want to consider.
Kepler’s conjecture is straightforward and appeals to common sense but it has landed mathematicians in a difficult philosophical debate. For nearly 500 years the conjecture remained unproved, until in 1998 the mathematician Thomas Hales announced that he had cracked it. The trouble was that his proof involved an exhaustive search that could only be completed by a computer. It involved over three gigabytes of computer code and data which made it humanly impossible to check the proof. Mathematicians have been quoted as saying they are “99% sure” that the proof is correct, but 99% is not good enough in mathematics. The question whether computer assisted proofs should be allowed remains a hot one in mathematics and, assuming that they are not, Kepler’s conjecture remains open.
But back to our kissing situation, we might consider a slightly different problem: imagine each person in turn kisses all the other people, regardless of whether they have kissed them before, and to top it off, they also kiss themselves. Then clearly for n people you get n2 kisses. Arranging the kisses on paper you simply get a square.
You can also get such a square by taking the triangle corresponding to the nth triangular number and then gluing to it, at its base, the triangle corresponding to the n-1st triangular number:
Which gives us the neat result that the nth square number is the sum of the nth and n-1st triangular numbers.
It all started with a pair of slightly unbelievable baby rabbits, a baby boy rabbit and a baby girl rabbit.
They were fully grown after one month
and, being rabbits, they did what rabbits do best, so that the next month two more baby rabbits (again a boy and a girl) were born.
The next month these babies were fully grown and the first pair had two more baby rabbits (again, handily a boy and a girl).
Ignoring problems of in-breeding, the next month the two adult pairs each have a pair of baby rabbits and the babies from last month mature.
This highly unrealistic situation (where rabbits never die and every month each adult pair produces a mixed pair of baby rabbits, who go on to mate the next month) was studied in the thirteenth century by Leonardo Pisano, better known as Fibonacci. He posed the problem “how many rabbits can a single pair produce after a year?”Fibonacci realised that the number of adult pairs in a given month was the total number of rabbits (both adults and babies) in the previous month
Adultsnth month = Rabbits(n-1)th month
and the number of baby pairs in a given month was the number of adult pairs in the previous month
Babiesnth month = Adults(n-1)th month = Rabbits(n-2)th month,
so the total number of pairs of rabbits in a particular month was the sum of the total pairs of rabbits in the previous two months:
Rabbitsnth month = Adultsnth month + Babiesnth month
= Rabbits(n-1)th month + Rabbits(n-2)th month.
So following this through for a year we get the answer to Fibonacci’s question: 144 pairs of rabbits after a year.
1 pair in January
1 pair in February
2 pairs in March
3 pairs in April
5 pairs in May
8 pairs in June
13 pairs in July
21 pairs in August
34 pairs in September
55 pairs in October
89 pairs in November
144 pairs in December
The first nine of these – 1, 1, 2, 3, 5, 8, 13, 21, 34 – are the sequence of numbers you see on your cube. The sequence bares Fibonacci’s name after he published it in his great work, the Liber Abaci, in 1202. This book changed the face of western mathematics, not because of the Fibonacci numbers, but because it introduced the now familiar numerals 0,1,2,…,9 and demonstrated how easy they made calculations, compared with the cumbersome Roman numerals that were still being used then by mathematicians and merchants.
The sequence has many curious mathematical features. For example, every second Fibonacci number, starting with 5, is the long side of a right-angled triangle whose side lengths are whole numbers: 5 is the longest side of a triangle whose other sides are 3 and 4 (52=32+42) and 13 is the longest side of a triangle whose other sides are 5 and 12 (132=52+122) and so on.
The Fibonacci sequence appears in many places in nature, for example we can find its numbers in the turns of a shell. Start with two squares side by side, each with sides of length 1, for the first two numbers in the sequence.
Then you can add a square with sides of length 2
and one with sides length 3
_ _ _ _ _
| | |
|_ _| |
|_|_| _ _ _|
and so on. And if we inscribe quarter circles in each of these squares they build up into a spiral shape, not a perfect mathematical spiral but one very close to some of the spirals we see in nature, such as in nautilus shells, that are built up compartment by compartment.
Nautilus Shell Cross-Section
Fibonacci numbers also appear in the spirals of seed heads on many plants. If you look closely at a sunflower you’ll notice the seeds appear in spirals, both clockwise and anticlockwise. And if you count the number of spirals in each direction, whether at the edge of the seedhead or closer in the middle, you are almost certain to find that they will be a pair of successive Fibonacci numbers.
Yellow Chamomile head showing the arrangement in 21 (blue) and 13 (aqua) spirals.
Such arrangements involving consecutive Fibonacci numbers appear in a wide variety of plants.
This is because the ratio of successive Fibonacci numbers, where each is divided by the number before it (1/1 = 1, 2/1 = 2, 3/2 = 1·5, 5/3 = 1·666…, 8/5 = 1·6, 13/8 = 1·625, 21/13 = 1·61538…), gradually approach a number known as the golden ratio. This number, approximately 1.618304, is an irrational number (it can’t be expressed as a simple fraction a/b where a and b are whole numbers), like pi and e on the other faces of your cube. Every irrational number can be better and better approximated by a sequence of fractions and mathematicians have found a way of measuring how well this approximation works. In this sense, the golden ratio is in fact the “worst approximable” number. This stems from the fact that you can represent it as the infinite continued fraction 1+1/(1+1/(1+1/(…) and is the reason behind the appearance of the golden ratio and Fibonacci numbers in plants.
Plants produce their leaves and seeds from a growth tip that spirals around the plant or centre of the seed head as it goes. If it turned by something close to a simple fraction of a turn the seeds would soon line up creating loosely packed spirals of seeds. The most efficient way to pack seeds into the seed head is for the number of turns between seeds to be an irrational number that is not easily approximated by a fraction, such as the golden ratio, avoiding the widely spaced spiralling arms. Instead you get the closely packed Fibonacci spirals of seeds, with the number of clockwise and anticlockwise spirals at any point on the seed head a pair of successive Fibonacci numbers, as these approximate the golden ratio.