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.
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) Exploration Guides and Paper 3 Resources
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.
IB Maths Resources from Intermathematics 
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October 2, 2018 at 11:59 pm
Graham G
There’s a serious mistake regarding orders of magnitude in this article.
It says “4×10^17 is a number so mind bogglingly big it would take about 45 trillion years to write out, writing 1 digit every second.” Not so, I can easily write it in this comment field: 400,000,000,000,000,000. The number of digits can be found by finding the power of 10 (in this case 17), and adding 1 (assuming the coefficient is less than 10).
Perhaps what was meant was that if every such number in the sequence of even numbers was written, it would take about 45 trillion years. Even then, that’s a bit too large. Calculating the number of total digits, one would have to write about 33,086,419,753,086,420 digits (A014925(16) * 5 * 4). This however would only take about 33,086,419,753,086,420 / (60 sec * 60 min * 24 hrs * 365.2425 yrs) ≈ 1,048,466,904 years to write the digits, at a rate of 1 digit per second. So in truth, it would only take about 1 billion years, which is 4 orders of magnitude difference from the 45 trillion estimate. This seems to suggest that either the original calculation used some unorthodox rounding, or that it was calculating something else entirely.
Link to the relevant OEIS sequence: http://oeis.org/A014925
October 3, 2018 at 7:52 am
Ibmathsresources.com
thanks – I have amended the post. It was a long time since I wrote this, looking back I think I was meaning the length of time to check each number.