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### Goldbach Conjecture Explained in 5 Minutes

Goldbach Conjecture Explained in 5 Minutes

In 1742, the German mathematician Christian Goldbach, in a discussion with the mathematician Leonhard Euler, made a simple statement..

Every even integer greater than 2 can be written as the sum of two prime numbers.

Mathematicians have tried to prove this ever since. None have. It's a great example of how a simple statement in mathematics can be amazingly difficult to prove. Computers have checked billions of numbers and shown it to be true for every number tested, but that's not the same as a proof. A proof would show it to be true for all even numbers, period.

It's easy to prove that every integer can be written as a product of primes. This is called the prime decomposition of an integer. So for any integer m we have.. m=(p1^s1)*(p2^s2)*......*(pn^sn)

where p1,p2,...,pn are prime numbers and the s1,s2,s3,...,sn are just integer powers and represent the simple fact that primes may be repeated. By collecting like primes together and raising them to a power we make sure that p1,p2,p3,...,pn are distinct primes with no duplication.

The condition that m be even simply means that one of these primes must be 2. Since the order of multiplication does not matter we can make p1=2. So m now looks like this..

m=(2^s1)*(p2^s2)*...*(pn^sn)

Goldbach's Conjecture says that when m is even there exists two prime numbers, let's call them g1 and g2, such that..

m=(2^s1)*(p2^s2)*...*(pn^sn)=g1+g2

Are we on our way to a proof? No, but it's still fun to try!

Perhaps there's some deep clue in the fact that Goldbach's Conjecture only works if p1=2. In other words, if 2 does not appear in the prime decomposition of an integer then Goldbach's Conjecture does not work. What's so special about the number 2?