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The idea of generating prime numbers through sequence of sets of co-primes was the starting point of this paper that ends up by proving two conjectures, the existence of infinitely many twin primes and the Goldbach conjecture. The main idea of our approach is summarized on the creation and on the analyzing sequence of sets of distinct co-primes with the first $n$ primes, $left{ p_i :, ileq n right}$, and the important properties of the modulus linear combination of the co-prime sets, $H=left(1,p_{n+1},..., Pi_{i=1}^n p_i-1right) $, that gives sets of even numbers ${0,2,4,..., Pi_{i=1}^n p_i -2 }$. Furthermore, by generalizing our approach, the Polignac conjecture the existence of infinitely many cousin primes, $p_{n+1}-p_{n}=4$, and the statement that every even integer can be expressed as a difference of two primes, are derived as well.
We use Maynards methods to show that there are bounded gaps between primes in the sequence ${lfloor nalpharfloor}$, where $alpha$ is an irrational number of finite type. In addition, given a superlinear function $f$ satisfying some properties describ
Let $t in mathbb{N}$, $eta >0$. Suppose that $x$ is a sufficiently large real number and $q$ is a natural number with $q leq x^{5/12-eta}$, $q$ not a multiple of the conductor of the exceptional character $chi^*$ (if it exists). Suppose further that,
We present in this work a heuristic expression for the density of prime numbers. Our expression leads to results which possesses approximately the same precision of the Riemanns function in the domain that goes from 2 to 1010 at least. Instead of usi
We present the pattern underlying some of the properties of natural numbers, using the framework of complex networks. The network used is a divisibility network in which each node has a fixed identity as one of the natural numbers and the connections
It is proven that there are infinitely prime pairs whose difference is no greater than 20.