Florentin Smarandache
This article presents a necessary and sufficient theorem for N numbers, coprime two by two, to be prime simultaneously. It generalizes V. Popa's theorem, as well as I. Cucurezeanu's theorem, Clement's theorem, S. Patrizio's theorems, etc. Particularly, this General Theorem offers many characterizations for twin primes, for quadruple primes, and so on.
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The theorem presented in this paper allows the creation of large prime numbers (of order up to o(n^2)) given a table of all primes up to n.
Using a sieve procedure akin to the sieve of Eratosthenes we show how for each prime $p$ to build the corresponding Möbius prime-function, which in the limit of infinitely large primes becomes identical to the original Möbius function. Discussing this limit we present two simple proofs of the Prime Number Theorem. In the framework of this approach we give several proofs of the infinitude of primes.
According to the similarity theorem on the distributions of the effective prime factors and by using two-part method, Goldbach theorem and, consequently, Goldbach conjecture was proved.
In this paper, we will give some estimation for the average error of the prime number theorem.
It is shown that the Mean Value Theorem for arithmetic functions, and simple properties of the zeta function are sufficient to assemble proofs of the Prime Number Theorem, and Dirichlet Theorem. These are among the simplest proofs of the asymptotic formulas of the corresponding prime counting functions.
Necessary condition for divisibility by the power of prime ideal and its application to Fermat's Last Theorem
该研究建立整元素被非分歧素理想素因子幂整除的必要条件，并据此为费马大定理提供简洁证明。
This 1964 paper developed as an off-shoot to the foundational query: Do we discover the natural numbers (Platonically), or do we construct them linguistically? The paper also assumes computational significance in the light of Agrawal, Kayal and Saxena's August 2000 paper, "PRIMES is in P", since both the TRIM and Compact Number Generating algorithms - each of which generates all the primes - are deterministic algorithms that run in polynomial time and suggest that the Prime Difference, d(n), is O(Number of primes < (p(n)^1/2)).
On Fermat's Last Theorem for odd prime exponents
本文证明了对于任意奇素数指数 $p$，费马方程 $x^p + y^p = z^p$ 不存在互素的正整数解。
Let $μ(n)$ denote the Möbius function, define $M(x)= \sum_{n\leq x}^{}μ(n)$. The main result of this paper is to prove that \begin{equation*} \displaystyle\lim_{x \to +\infty}\frac{M(x)}{x}=0 \end{equation*} which is equivalent to the prime number theorem. We also use Selberg's asymptotic formula, but the treatments of key parts are different from several classical proofs.
Twin prime number problem is mainly the structure of the twin prime numbers and whether there are infinitely many prime twins group. In this paper, by constructing a special cluster number set(see formula(2.3)in the paper), proves that the number of set number of the first n columns set the intersection of the minimum number of q is decision of the prime twins (q, q+2), and the minimum number of series is divergent(see Theorem 2).The main rezults are Theorem 2,Theorem 3 and Theorem 4.Prime twins so thoroughly proved there must be infinitely many groups of twin prime conjecture.
1. There is no existing any quadratic interval $η_{n}:=(n^{2},(n+1)^{2}],$ which contains less than 2 prime numbers. The number of prime numbers within $η_{n}$ goes averagely linear with n to infinity. 2. The exact law of the number $π(n)$ of prime numbers smaller or equal to n is given. As an approximation of that we get the prime number theorem of Gauss for great values of n. 3. We derive partition laws for $π(η_{n})$, for the number of twin primes $π_{2}(η_{n})$ in quadratic intervals $η_{n}$ and for the multiplicity $π_{g}(2n)$ of representations of Goldbach-pairs for a given even number 2…
In scientific paper, we will show a proof of Legendre's conjecture based on a scheme for finding elements of "active" set ${\rm H}_{m^{4}}$ and "critical" element $C_{m^{4}}$ for number $m^{4}$ at each $m \ge 3$.
We prove dual theorems to theorems proved by author in \cite {5}. Beginning with Section 10, we introduce and study so-called "twin numbers of the second kind" and a postulate for them. We give two proofs of the infinity of these numbers and a sufficient condition for truth of the postulate; also we pose several other conjectures. Finally, we consider a conception of axiom of type "AiB".
In this paper we use Dirichlet's theorem in order to elementally prove two theorems. The first says that since a polynomial ax+b generates one prime, it also generates infinites. The second theorem (which is proved in a very simillar way to the first) says that x^2+1 generates infinitely many primes.
This paper is devoted to the theory of prime numbers. In this paper we first introduce the notion of a matrix of prime numbers. Then, in order to investigate the density of prime numbers in separate rows of the matrix under consideration, we propose a number of lemmas and theorems that, together with the Dirichlet and Euler theorems, make it possible to prove the infinity of prime twins.
We show that for any $P= 6^{m+1}.N -1 $ is a prime number for any $1 < N \le 13$ , $N \ne 8$ and $N \ne i^{m+1}Mod(6i+1) $ where $ i \in Z^+ $ and $ m \in $ $odd$ $Z^+ $ for $1 < N \le 13$ and $N \ne 8$ and also we further discussed that $P= 6^{m+1}.N -1 $ is a prime number for $ N >13 $ if and only if , $N \ne i^{m+1}Mod(6i+1) +(6i +1)a $ $ ; i,a \le Z^+ $
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