prime array 4

How Are Prime Numbers Distributed? Twin Primes Conjecture

Thanks to a great post on the Teaching Mathematics blog about getting students to conduct an open ended investigation on consecutive numbers, I tried this with my year 10s – with some really interesting results. My favourites were these conjectures:

1) In a set of any 10 consecutive numbers, there will be no more than 5 primes. (And the only set of 5 primes is 2,3,5,7,11)
2) There is only 1 example of 3 consecutive odd numbers all being primes – 3,5,7

(You can prove both in a relatively straightforward manner by considering that a span of 3 consecutive odd numbers will always contain a multiple of 3)

Twin Prime Conjecture

These are particularly interesting because the study of the distribution of prime numbers is very much a live mathematical topic that mathematicians still work on today. Indeed studying the distribution of primes and trying to prove the twin prime conjecture are important areas of research in number theory.

The twin prime conjecture is one of those nice mathematical problems (like Fermat’s Last Theorem) which is very easy to understand and explain:

It is conjectured that there are infinitely many twin primes – ie. pairs of prime numbers which are 2 away from each other. For example 3 and 5 are twin primes, as are 11 and 13. Whilst it is easy to state the problem it is very difficult to prove.


However, this year there has been a major breakthrough in the quest to answer this problem. Chinese mathematician Yitang Zhang has proved that there are infinitely many prime pairs with gap N for some N less than 70,000,000.

This may at first glance not seem very impressive – after all to prove the conjecture we need to prove there are infinitely many prime pairs with gap N = 2. 70,000,000 is a long way away! Nevertheless this mathematical method gives a building block for other mathematicians to tighten this bound. Already that bound has been reduced to N <60,744 and is being reduced almost daily.

Prime Number Distribution

Associated with research into twin primes is also a desire to understand the distribution of prime numbers. Wolfram have a nice demonstration showing the cumulative distribution of prime numbers (x axis shows total integers x100)

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Indeed, if you choose at random an integer from the first N numbers, the probability that it is prime is approximately given by 1/ln(N).

We can see other patterns by looking at prime arrays:

prime array

This array is for the first 100 integers – counting from top left to right.  Each black square represents a prime number.  The array below shows the first 5000 integers.  We can see that prime numbers start to “thin out” as the numbers get larger.

primae array 2The desire to understand the distribution of the prime numbers is intimately tied up with the Riemann Hypothesis – which is one of the million dollar maths problems.  Despite being conjectured by Bernhard Riemann over 150 years ago it has still to be proven and so remains one of the most important unanswered questions in pure mathematics.

For more reading on twin primes and Yitang Zhang’s discovery, there is a great (and detailed) article in Wired on this topic.

If you enjoyed this topic, you may also like:

A post on synesthesia about how some people see colours in their numbers.

A discussion about the Million Dollar Maths problems (which includes the Riemann Hypothesis).

IB Revision

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If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

Screen Shot 2018-03-19 at 4.42.05 PM.pngThe Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

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The Practice Exams section takes you to ready made exams on each topic – again with worked solutions.  This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.