Additive properties of prime numbers
In June, I will present about twin prime numbers as I wrote before. I will talk some properties about “prime numbers” at first.
As you know, there exist infinitely many prime numbers. This can be proven easily. If there are only finitely many ones, the product of all of them is a integer. The number that is made by it added to 1 can’t be divided by any prime numbers. It contradicts the hypothesis. This is a famous proof.
However another proof seems more interesting: the sum of inverses of all prime numbers diverges, so there exist infinitely many ones. A zeta function is used for calculating the sum. The prime numbers theory needn’t be used.
One of purposes of the presentation is convergence of the sum of inverses of all twin prime numbers. This contrast between prime numbers and twin prime numbers seems weird. This indicates that the density of twin prime numbers is much lower than that of prime numbers even if the twin prime conjecture is right.