Abstract: We prove some results concerning the distribution of primes assuming the Riemann hypothesis. First, we prove the explicit result that there exists a prime in the interval $(x-\frac{4}{\pi} \sqrt{x}...

Abstract: Euclid’s proof that the prime numbers are “more than any assigned multitude” (Elements, proposition IX, 20) has long been hailed as a model of elegance and simplicity. Yet, surprisingly, it has also been misrepresented in a great many accounts: The article Hardy and Woodgold (2009) gives a detailed list of sources, including many by eminent number theorists, that either erroneously describe the structure of Euclid’s proof or make false historical claims about it. It is wise, therefore, to begin by quoting Euclid’s argument directly, as it is given in Heath’s translation (Heath 1956, vol. II, p. 412).

Abstract: Divisibility of Integers The Concept of Divisibility The Greatest Common Divisor and The Least Common Multiple The Euclidean Algorithm Solving Linear Diophantine Equations Prime Factorization of Integers Congruences Residue Classes and Systems of Residues Euler's Theorem Wilson's Theorem Congruence Equations Basic Concepts of Congruences of High Degrees Linear Congruences Systems of Linear Congruence Equations and the Chinese Remainder Theorem General Congruence Equations Quadratic Residues The Legendre Symbol and the Jacobi Symbol Exponents and Primitive Roots Exponents and Their Properties Primitive Roots and Their Properties Indices, Construction of Reduced System of Residues Nth Power Residues Some Elementary Results for Prime Distribution Introduction to the Basic Properties of Primes and The Main Results of Prime Number Distribution Proof of the Euler Product Formula Proof of a Weaker Version of the Prime Number Theorem Equivalent Statements of the Prime Number Theorem Simple Continued Fractions Simple Continued Fractions and Their Basic Properties Simple Continued Fraction Representations of Real Numbers Application of Continued Fraction In Cryptography-Attack to RSA with Small Decryption Exponents Basic Concepts Maps Algebraic Operations Homomorphisms and Isomorphisms between Sets with Operations Equivalence Relations and Partitions Group Theory Definitions Cyclic Groups Subgroups and Cosets Fundamental Homomorphism Theorem Concrete Examples of Finite Groups Rings and Fields Definition of a Ring Integral Domains, Fields, and Division Rings Subrings, Ideals, and Ring Homomorphisms Chinese Remainder Theorem Euclidean Rings Finite Fields Field of Fractions Some Mathematical Problems in Public Key Cryptography Time Estimation and Complexity of Algorithms Integer Factorization Problem Primality Tests The RSA Problem and the Strong RSA Problem Quadratic Residues The Discrete Logarithm Problem Basics of Lattices Basic Concepts Shortest Vector Problem Lattice Basis Reduction Algorithm Applications of LLL Algorithm References Further Reading Index

Abstract: On the ABC conjecture, we get an asymptotic estimate for the number of squarefree values of a polynomial at prime arguments. A key tool in our argument is a result by Tao and Ziegler (improving a previous result by Green and Tao) concerning arithmetic progressions of primes.

Abstract: Several examples of generalized number systems are constructed to compare various conditions occurring in the literature for the prime number theorem in the context of Beurling generalized primes.

Abstract: The enlargement of known zero-free regions has enabled us to find better effective estimates for classical number-theoretic functions linked to the distribution of prime numbers. In particular we draw the quintessence of the method of Rosser and Schoenfeld on the upper bounds for the usual Chebyshev prime and prime power counting functions to find an upper bound function directly linked to a zero-free region.

Abstract: The aim of this work is to estimate exponential sums of the form Σn≤xΛ(n) exp(2iπ(f(n) + βn)), where Λ denotes von Mangoldt's function, f a digital function, and β ∈ double-struck R a parameter. This result can be interpreted as a Prime Number Theorem for rotations (i.e. a Vinogradov type theorem) twisted by digital functions.

Abstract: Suppose f(x) is a monic irreducible polynomial with integer coefficients. If p is a prime number, then reducing the coefficients of f(x) modulo p gives a new polynomial fp(x), which may be reducible. We say that f(x) is split modulo p if fp(x) is the product of distinct linear factors. Example 1.1.1. The polynomial f(x) = x2 + 1 is split modulo 5, since f5(x) ≡ (x + 2)(x + 3) mod 5. But it is not split modulo 7, since f7(x) is irreducible, nor is it split modulo 2, since f2(x) ≡ (x + 1)2 (mod 2) has a repeated factor. The first few p for which x2 + 1 is split modulo p are 5, 13, 17, 29, 37, 41, 53, . . . .

Abstract: In recent years, Chinese remainder theorem (CRT)-based function sharing schemes are proposed in the literature. In this paper, we study systems of two or more linear congruences. When the moduli are pairwise coprime, the main theorem is known as the CRT, because special cases of the theorem were known to the ancient Chinese. In modern algebra the CRT is a powerful tool in a variety of applications, such as cryptography, error control coding, fault-tolerant systems and certain aspects of signal processing. Threshold schemes enable a group of users to share a secret by providing each user with a share. The scheme has a threshold t+1 if any subset with cardinality t+1 of the shares enables the secret to be recovered. In this paper, we are considering 2t prime numbers to construct t share holders. Using the t share holders, we split the secret S into t parts and all the t shares are needed to reconstruct the secret using CRT.

Abstract: A lower bound for the p-adic valuation of the number Ep = Σn=1∞n! ∈ ℚp defined by an Euler-type series is proved in the paper for infinitely many prime numbers p.

Abstract: We show that arithmetic local constants attached by Mazur and Rubin to pairs of self-dual Galois representations which are congruent modulo a prime number $p>2$ are compatible with the usual local constants at all primes not dividing $p$ and in two special cases also at primes dividing $p$ . We deduce new cases of the $p$ -parity conjecture for Selmer groups of abelian varieties with real multiplication (Theorem 4.14) and elliptic curves (Theorem 5.10).

Abstract: We consider Linnik’s type of the Waring–Goldbach problem with unequal powers of primes. In particular, it is proved that every sufficiently large even integer can be represented as a sum of one prime, one square of prime, one cube of prime, one fourth power of prime and 18 powers of 2.

Abstract: We estimate from below the lower density of the set of prime numbers $p$ such that $p-1$ has a prime factor of size at least $p^c$, where $1/4 \le c \leq 1/2$. We also establish upper and lower bounds on the counting function of the set of positive integers $n\le x$ with exactly $k$ prime factors, counted with or without multiplicity, such that the largest prime factor of ${\text{\rm gcd}}(p-1: p\mid n)$ exceeds $n^{1/2k}$.

Abstract: For each prime number p, the dynamical behavior of the square mapping on the ring Zp of p-adic integers is studied. For p = 2, there are only attracting fixed points with their attracting basins. For p 3, there are a fixed point 0 with its attracting basin, finitely many periodic points around which there are countably many minimal components and some balls of radius 1=p being attracting basins. All these minimal components are precisely exhibited for different primes p.

Abstract: This paper presents a modified Paley method for calculation of Mersenne matrices at order values equal to odd prime numbers Some examples of Mersenne matrix sorting, allowing for calculation of the complete set of functions, are also considered A comparison of Walsh and Mersenne-Walsh systems of functions in terms of their properties and fields of application is provided The efficiency of this topic for use in the development of band-pass filters is indicated

Abstract: We establish the first pointwise ergodic theorems along thin sets of prime numbers; a set with zero density with respect to the primes. For instance we will be able to achieve this with the Piatetski–Shapiro primes. Our methods will be robust enough to solve the ternary Goldbach problem for some thin sets of primes.

Abstract: Let p be a prime number and F a field containing a root of unity of order p. We relate recent results on vanishing of triple Massey products in the mod-p Galois cohomology of F, due to Hopkins, Wickelgren, Minayc, and Tan, to classical results in the theory of central simple algebras. For global fields, we prove a stronger form of the vanishing property.

Abstract: The security of RSA algorithm depends upon the positive integer N, which is the multiple of two precise large prime numbers. Factorization of such great numbers is a problematic process. There are many algorithms has been implemented in the past years. The offered KNJ -Factorization algorithm contributes a deterministic way to factorize RSA. The algorithm limits the search by only considering the prime values. Subsequently prime numbers are odd numbers accordingly it also requires smaller number steps to factorize RSA. In this paper, the anticipated algorithm is very simple besides it is very easy to understand and implement. The main concept of this KNJ factorization algorithm is, to check only those factors which are odd and prime. The proposed KNJ- Factorization algorithm works very efficiently on those factors; which are adjoining and close to N. The proposed factorization method can speed up if we can reduce the time for primality testing. It fundamentally decreases the time complexity.

Abstract: There are 4 prime numbers less than 10; there are 25 primes less than 100; there are 168 primes less than 1000, and 1229 primes less than 10000. At what rate do the primes thin out? Today we use the notation π(x) to denote the number of primes less than or equal to x; so π(1000) = 168.

Abstract: . In a 1737 paper, Euler gave the first proof that the sum of the reciprocals of the prime numbers diverges. That paper can be considered as the founding document of analytic number theory, and its key innovation — socalled Euler products — are now ubiquitous in the field. In this note, we probe Euler’s claim there that “the sum of the reciprocals of the prime numbers” is “as the logarithm” of the sum of the harmonic series. Euler’s argument for this assertion falls far short of modern standards of rigor. Here we show how to arrange his ideas to prove the more precise claim that X

Abstract: We produce an upper bound on the number of extended irreducible binary quartic Goppa codes of length 2 n +1, where n > 3 is a prime number.

Abstract: The irregular distribution of prime numbers amongst the integers has found multiple uses, from engineering applications of cryptography to quantum theory. The degree to which this distribution can be predicted thus has become a subject of current interest. Here, we present a computational analysis of the deviations between the actual positions of the prime numbers and their predicted positions from Riemann’s counting formula, focused on the variance function of these deviations from sequential enumerative bins. We show empirically that these deviations can be described by a class of probabilistic models known as the Tweedie exponential dispersion models that are characterized by a power law relationship between the variance and the mean, known by biologists as Taylor’s power law and by engineers as fluctuation scaling. This power law behavior of the prime number deviations is remarkable in that the same behavior has been found within the distribution of genes and single nucleotide polymorphisms (SNPs) within the human genome, the distribution of animals and plants within their habitats, as well as within many other biological and physical processes. We explain the common features of this behavior through a statistical convergence effect related to the central limit theorem that also generates 1/f noise.

Abstract: For any positive integer k, we show that infinitely often, perfect kth powers appear inside very long gaps between consecutive prime numbers, that is, gaps of size $$\displaystyle{c_{k}\frac{\log p\log _{2}p\log _{4}p} {(\log _{3}p)^{2}},}$$ where p is the smaller of the two primes.

Abstract: RSA is one the most well-known public-key cryptosystems widely used for secure data transfer. An RSA encryption key includes a modulus n which is the product of two large prime numbers p and q. If an RSA modulus n can be decomposed into p and q, the corresponding decryption key can be computed easily from them and the original message can be obtained using it. RSA cryptosystem relies on the hardness of factorization of RSA modulus. Suppose that we have a lot of encryption keys collected from the Web. If some of them are inappropriately generated so that they share the same prime number, then they can be decomposed by computing their GCD (Greatest Common Divisor). Actually, a previously published investigation showed that a certain ratio of RSA moduli in encryption keys in the Web are sharing prime numbers. We may find such weak RSA moduli n by computing the GCD of many pairs of RSA moduli. The main contribution of this paper is to present a new Euclidean algorithm for computing the GCD of all pairs of encryption moduli. The idea of our new Euclidean algorithm that we call Approximate Euclidean algorithm is to compute an approximation of quotient by just one 64-bit division and to use it for reducing the number of iterations of the Euclidean algorithm. We also present an implementation of Approximate Euclidean algorithm optimized for CUDA-enabled GPUs. The experimental results show that our implementation for 1024-bit GCD on GeForce GTX 780Ti runs more than 80 times faster than the Intel Xeon CPU implementation. Further, our GPU implementation is more than 9 times faster than the best known published GCD computation using the same generation GPU.

Abstract: Let E/Q be an elliptic curve and p be a prime number, and let G be the Galois group of the extension of Q obtained by adjoining the coordinates of the p-torsion points on E. We determine all cases when the Galois cohomology group H^1(G, E[p]) does not vanish, and investigate the analogous question for E[p^i] when i>1. We include an application to the verification of certain cases of the Birch and Swinnerton-Dyer conjecture, and another application to the Grunwald-Wang problem for elliptic curves.