Lesson 12

Finally, here are some examples of open problems in number theory (that is, questions to which no one knows the answer yet).

• Goldbach's Conjecture.  Does there exist an even number greater than two that cannot be expressed as the sum of two primes?


• Collatz Conjecture.  For a number \(n \in \mathbb{Z}_{>0}\), consider the following sequence: if \(n\) is even, replace it with \(n/2\), and if it is odd, replace it with \(3n+1\), and then continue the same process. Does there exist a number \(n \in \mathbb{Z}_{>0}\) for which this sequence does not encounter one?
  

  



• Twin Primes.  Does there exist an infinite number of pairs of prime numbers that differ by two?


• Perfect Numbers.  Does there exist an odd number that is equal to the sum of its proper positive divisors?