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.

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.