I suggest a probabilistic approach that helps to address some classical questions and problems of Number Theory, like the Goldbach Conjecture [1], distributions of twin- and d-primes, prime numbers among arithmetic sequences and others. The concepts of ‘randomness’ and ‘independence’ relevant to number-theoretic problems are discussed here, and the basic concepts of divisibility of natural number are interpreted in terms of probability spaces and appropriate probability distributions on classes of congruence. I analyze and demonstrate the importance of Zeta probability distribution and prove theorems stating the equivalence of probabilistic independence of divisibility of random integers by coprime factors, and the fact that random variables with the property of independence of coprime factors must have Zeta probability distribution. The idea to use Zeta distribution is motivated by the fact that it provides the validity of the probabilistic Cramér’s model for asymptotic prime number distribution, in full agreement with the Prime Number Theorem. Multiplicative and additive models with recurrent equations for generating sequences of prime numbers are derived based on the reduced Sieve of Eratosthenes Algorithm. This allows to interpret such sequences as realizations of random walks on set of natural numbers and on multiplicative semigroups generated by sets of all prime numbers, representing paths of stochastic dynamical systems. The H. Cramér’s model for probability distribution of primes is modified as a generalized predictable non-stationary Bernoulli process with unequally distributed terms that are asymptotically pairwise independent. This model is applied to analyze the sequences of primes generated by appropriate random walks. With intense use of Zeta probability distribution, it seems possible by using the modified Cramér’s model to approximate probability distribution of various arithmetic function. Since probabilistic approach meets certain skepticism and even disbelief from a part of mathematicians working in traditional manner in Number Theory, I decided to attack the problem of Strong Goldbach Conjecture (SGC) from pure deterministic point of view. As a result, I derived a recursive formula which generates a sequence {G(m)} of consecutive nonempty Goldbach sets. Each Goldbach set G(m) is asset of all prime numbers solving equation p + p’ = 2m for any natural number m > 2. The recursive formula justifies SGC by mathematical induction. Thus, this work represents two independent proofs of validity for Strong Goldbach Conjecture.