Proposed Proofs of Goldbach's Conjecture et al.

The conjectures of Andrica and Legendre are strongly

related to the gap between two consecutively prime numbers.

We prove Andrica's conjecture using the Mean Value Theorem

for a continuous and concave function. Then we show that

Legendre's conjecture is a consequence of Andrica's conjecture.

In his famous article of 1859 Riemann obtained

a formula for the number of primes less than a real number

expressed in terms of the zero's of the zeta function.

Von Mangoldt proved that this formula is valid. As a consequence, the non-trivial zeros must line up. We show that all the non-trivial zeros of the zeta function are on the critical line.

The conjecture of Goldbach is the best known unsolved mathematical problem and is very hard to prove. We assume that a possible proof will show a strong relationship between primes and positive even numbers.

Additionally, we solve the conjecture of Polignac using Goldbach and Euclid's proof.