At the start of The Impossible Clue, before she’s caught up in the case of a disappearing scientist and a missing invisibility suit, Alice Jones is working on a much more mathematical mystery.
“I was trying to prove Goldbach’s Conjecture, but I wasn’t getting very far. Maybe it was the fact that it hadn’t been cracked by some of the greatest mathematical brains of the twentieth century. Then again, maybe it was the steady stream of spitballs hitting the side of my head that was stopping me.”
Goldbach’s Conjecture is one of the most well-known unsolved mathematical mysteries. It states that every integer greater than 2 can be expressed as the sum of two primes.
Boiled down that means that every whole number (no fractions or decimals please) bigger than 2 can me made by adding two prime numbers together.
Prime numbers are numbers that are only divisible by 1 and themselves. The first 10 primes are: 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. There are an infinite number of prime numbers – but as they get larger, they’re a lot harder to calculate!
The largest known prime number is 274,207,281 − 1. That’s a number with 22,338,618 digits! If I was to type out the whole thing on A4 paper, it would be 4,647 pages long (depending on your font-I chose Ariel 11pt)!
Mathematicians have been trying to prove this theory since Christian Goldbach first suggested it in 1742, but none of them have been able to prove that it’s true.
Because so far the only way to know if a number can be made by adding two prime numbers together, is to check it by hand. Even if you wrote a computer programme that could check each number for you, there are an infinite number of numbers, so you would never know for sure. What Alice and other mathematicians are trying to find is a general proof: something that proves the conjecture must be true without having to check every single number to be sure.
For example here’s a (very) general proof of the Pythagorean Theorum (proper mathematicians will spot that this isn’t a formal proof-but it shows how a proof proves that if x is true in one case it must be true for all similar cases. If you want to see what a real formal proof looks like, click here.) :
Since the proof just uses letters to represent the sides of the triangle, not actual numbers, you can apply this proof to ALL right triangles and therefore a2 + b2 = c2 is always true.
No one has been able to do that for Goldbach’s conjecture. Though a lot of people have tried. Right now it has been proved to be true for n ≤ 4 × 1018 (all numbers greater than or equal to 4 up to 10 to the eighteenth power).
So will Alice be the one to come up with a general proof to the problem that’s stumped some of the greatest minds in the 20th (and 21st) century? Maybe. She’s only twelve, and she’s planning to keep studying maths. By the time she’s got her PhD, she might be the one to do it. She’ll certainly give it her best shot. Or maybe one of you will get there first?
If you are interested in getting involved in solving some mathematical mysteries of your own, check out The Great Internet Mersenne Prime Search. You can help find the next largest known prime number. There’s even a 3000 dollar prize!