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goldbach

The Goldbach Conjecture is one of the most famous problems in mathematics.  It has remained unsolved for over 250 years – after being proposed by German mathematician Christian Goldbach in 1742.  Anyone who could provide a proof would certainly go down in history as one of the true great mathematicians.  The conjecture itself is deceptively simple:

“Every even integer greater than 2 can be written as the sum of 2 prime numbers.”

It’s easy enough to choose some values and see that it appears to be true:

4: 2+2
6: 3+3
8: 3+5
10: 3+7 or 5+5

But unfortunately that’s not enough to prove it’s true – after all, how do we know the next number can also be written as 2 primes?  The only way to prove the conjecture using this method would be to check every even number.  Unfortunately there’s an infinite number of these!

Super-fast computers have now checked all the first 4×1017 even numbers  4×1017 is a number so mind bogglingly big it would take about 13 billion years to check all these numbers, checking one number every second. (4×1017)/(60x60x24x365) = 1.3 x 1010.   So far they have found that every single even number greater than 2 can indeed be written as the sum of 2 primes.

So, if this doesn’t constitute a proof, then what might?  Well, mathematicians have noticed that the greater the even number, the more likely it will have different prime sums.  For example 10 can be written as either 3+7 or 5+5.  As the even numbers get larger, they can be written with larger combinations of primes.  The graph at the top of the page shows this.  The x axis plots the even numbers, and the y axis plots the number of different ways of making those even numbers with primes.  As the even numbers get larger, the cone widens – showing ever more possible combinations.  That would suggest that the conjecture gets ever more likely to be true as the even numbers get larger.

goldbach2

A similar problem from Number Theory (the study of whole numbers) was proposed by legendary mathematician Fermat in the 1600s. He was interested in the links between numbers and geometry – and noticed some interesting patterns between triangular numbers, square numbers and pentagonal numbers:

Every integer (whole number) is either a triangular number or a sum of 2 or 3 triangular numbers.  Every integer is a square number or a sum of 2, 3 or 4 square numbers.  Every integer is a pentagonal number or a sum of 2, 3, 4 or 5 pentagonal numbers.

There are lots of things to investigate with this.  Does this pattern continue with hexagonal numbers?  Can you find a formula for triangular numbers or pentagonal numbers?  Why does this relationship hold?

If you like this post you might also like:

How Are Prime Numbers Distributed? Twin Primes Conjecture. Discussion on studying prime numbers – in particular the conjecture that there are infinitely many twin primes.

Fermat’s Last Theorem An introduction to one of the greatest popular puzzles in maths history.

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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IB Maths Exploration Guide

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I have just made a Udemy online tutorial course for the exploration.  This includes nine tutorial videos of essential information designed to ensure you get the best possible grade.

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IB HL Paper 3 Practice Questions (120 page pdf)

Eight P3 investigation questions and fully typed mark scheme (around 240 marks)

Available to download here

Modelling Guide for Explorations

A 50 page pdf guide full of advice to help with modelling explorations – focusing in on non-calculator methods in order to show good understanding.

Available to download here.

Statistics Guide

A 55 page pdf guide full of advice to help with modelling explorations – focusing in on non-calculator methods in order to show good understanding.

Available to download here.

IB Revision Notes

Full revision notes for SL Analysis (60 pages), HL Analysis (112 pages) and SL Applications (53 pages).  Beautifully written by an experienced IB Mathematics teacher.

Available to download here.

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