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 4×10^{17} even numbers (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). 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.

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