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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.

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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)

prime array 3

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).

Essential resources for IB students:

1) Revision Village

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Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

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There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank 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 a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

2) Exploration Guides and Paper 3 Resources

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I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

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All content on this site has been written by Andrew Chambers (MSc. Mathematics, IB Mathematics Examiner).

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P3 investigation questions and fully typed mark scheme.  Packs for both Applications students and Analysis students.

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