Question

Let's prove the Goldbach Conjecture!

Answer 1

So this is the Goldbach Conjecture:

Every even number is the sum of 2 prime numbers.

Answer 2

-.-

Answer 3

Come on it'll be fun! XD

Answer 4

what about 2? that is even

Answer 5

lol keep going :D

Answer 6

Oh, it only applies for numbers greater than 2.

Answer 7

It has been shown by extreme brute force computer calculations to be true for millions of numbers.

Answer 8

also it implies that each odd greater than 3 can be written as sum of 3 primes

Answer 9

I've made a bit of progress on it since this afternoon when ganeshie mentioned it, I've been hooked ever since. So here goes with what I've come up with:

Answer 10

oh isee , show what u got

Answer 11

So the goldbach conjecture is saying \[\Large p+q=2n\] that this is true for all positive integers n, except for n=1. We can equivalently say, \[\Large \frac{p+q}{2}=n\] So what I am looking at is now saying that every integer greater than 1 is the average of two primes.

Answer 12

4

Answer 13

So now I'm going to look at averages. The only way you can write something that averages to a number is if both numbers are symmetrically away from it, so we can say: \[\Large \frac{n+n}{2}=n\] That would be one way of saying it, but let's push each one further away. \[\Large \frac{(n-1)+(n+1)}{2}=n \\ \Large \frac{(n-2)+(n+2)}{2}=n\\ ... \\ \Large \frac{(n-k)+(n+k)}{2}=n\]

Answer 14

So obviously when n is prime itself, it's true, but after that we need to start looking at the two adjacent primes, then move out layer by layer equally. So like for 8, we have: So they're symmetrical like this. Now my way of going about showing that the GC is true is by contradiction...