Abstract: This paper introduces new techniques to generate provable prime numbers efficiently on embedded devices such as smartcards, based on variants of Pocklington's and the Brillhart-Lehmer-Selfridge-Tuckerman-Wagstaff theorems. We introduce two new generators that, combined with cryptoprocessor-specific optimizations, open the way to efficient and tamper-resistant on-board generation of provable primes. We also report practical results from our implementations. Both our theoretical and experimental results show that constructive methods can generate provable primes essentially as efficiently as state-of-the-art generators for probable primes based on Fermat and Miller-Rabin pseudo-tests. We evaluate the output entropy of our two generators and provide techniques to ensure a high level of resistance against physical attacks. This paper intends to provide practitioners with the first practical solutions for fast and secure generation of provable primes in embedded security devices.

Abstract: A subset W of vertices of a graph G is called a resolving set for G if for every pair of distinct vertices u and v of G, there exists a vertex w∈W such that the distance between u and w is different from the distance between v and w. A resolving set containing a minimum number of vertices is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G, denoted by β(G). A subset S of vertices of a graph G is called a determining set if whenever two automorphisms agree on the elements of S, they agree on all of G. The minimum cardinality of a determining set of G is called the determining number of G, denoted by Det(G). In this paper, we find the metric dimension of Cayley graphs, Cay(Zn: S) for all n≥7 and S = {±1,±3}. Also we show that, for all prime numbers n = 2p+1 with p prime and any subset S of n Z \{0} with S = -S, S ≠φ and n SZ \{0} ≠ , n Det(Cay(Z :S))=2 .

Abstract: Let E be an elliptic curve over Q without complex multiplication. For each prime p of good reduction, let |E(F p)| be the order of the group of points of the reduced curve over F p. According to a conjecture of Koblitz, there should be infinitely many such primes p such that |E(F p)| is prime, unless there are some local obstructions predicted by the conjecture. Suppose that E is a curve without local obstructions (which is the case for most elliptic curves over Q). We prove in this paper that, under the GRH, there are at least 2.778C twin E x/(log x) 2 primes p such that |E(F p)| has at most 8 prime factors, counted with multiplicity. This improves previous results of Steuding & Weng [20, 21] and Miri & Murty [15]. This is also the first result where the dependence on the conjectural constant C twin E appearing in Koblitz's conjecture (also called the twin prime conjecture for elliptic curves) is made explicit. This is achieved by sieving a slightly different sequence than the one of [20] and [15]. By sieving the same sequence and using Selberg's linear sieve, we can also improve the constant of Zywina [24] appearing in the upper bound for the number of primes p such that |E(F p)| is prime. Finally, we remark that our results still hold under an hypothesis weaker than the GRH.

Abstract: Chebyshev observed in a letter to Fuss that there tends to be more primes of the form $4n+3$ than of the form $4n+1$. The general phenomenon, which is referred to as Chebyshev's bias, is that primes tend to be biased in their distribution among the different residue classes $\bmod q$. It is known that this phenomenon has a strong relation with the low-lying zeros of the associated $L$-functions, that is if these $L$-functions have zeros close to the real line, then it will result in a lower bias. According to this principle one might believe that the most biased prime number race we will ever find is the Li$(x)$ versus $\pi(x)$ race, since the Riemann zeta function is the $L$-function of rank one having the highest first zero. This race has density 0.99999973..., and we study the question of whether this is the highest possible density. We will show that it is not the case, in fact there exists prime number races whose density can be arbitrarily close to 1. An example of race whose density exceeds the above number is the race between quadratic residues and non-residues modulo 4849845, for which the density is 0.999999928... We also give fairly general criteria to decide whether a prime number race is highly biased or not. Our main result depends on the General Riemann Hypothesis and on a hypothesis on the multiplicity of the zeros of a certain Dedekind zeta function. We also derive more precise results under a linear independence hypothesis.

Abstract: Given an elliptic curve $E$ over a number field $K$, the $\ell$-torsion points $E[\ell]$ of $E$ define a Galois representation $\gal(\bar{K}/K) \to \gl_2(\ff_\ell)$. A famous theorem of Serre states that as long as $E$ has no Complex Multiplication (CM), the map $\gal(\bar{K}/K) \to \gl_2(\ff_\ell)$ is surjective for all but finitely many $\ell$. We say that a prime number $\ell$ is exceptional (relative to the pair $(E,K)$) if this map is not surjective. Here we give a new bound on the largest exceptional prime, as well as on the product of all exceptional primes of $E$. We show in particular that conditionally on the Generalized Riemann Hypothesis (GRH), the largest exceptional prime of an elliptic curve $E$ without CM is no larger than a constant (depending on $K$) times $\log N_E$, where $N_E$ is the absolute value of the norm of the conductor. This answers affirmatively a question of Serre.

Abstract: Let α ∈ ℤ\{0}. A positive integer N is said to be an α-Korselt number (Kα-number, for short) if N ≠ α and N - α is a multiple of p - α for each prime divisor p of N. By the Korselt set of N, we mean the set of all α ∈ ℤ\{0} such that N is a Kα-number; this set will be denoted by . Given a squarefree composite number, it is not easy to provide its Korselt set and Korselt weight both theoretically and computationally. The simplest kind of squarefree composite number is the product of two distinct prime numbers. Even for this kind of numbers, the Korselt set is far from being characterized. Let p, q be two distinct prime numbers. This paper sheds some light on .

Abstract: It is shown that any primitive integral Apollonian circle packing captures a fraction of the prime numbers. Basically, the method consists of applying the circle method and considering the curvatures produced by a well-chosen family of binary quadratic forms.

Abstract: In this paper, we propose a new primality test, and then we employ this test to find a formula for π that computes the number of primes within any interval. We finally propose a new formula that computes the nth prime number as well as the next prime for any given number.

Abstract: We give a representation of the classical theory of multiplicative arithmetic functions (MF) in the ring of symmetric polynomials written on the isobaric basis. The representing elements are recursive sequences of Schur-hook polynomials evaluated on subrings of the complex numbers. The multiplicative arithmetic functions are units in the Dirichlet ring of arithmetic functions, and their properties can be described locally, that is, at each prime number p. Our representation is, hence, a local representation. This representation enables us to clarify and generalize classical results, e.g., the Busche-Ramanujan identity, as well as to give a richer structural description of the convolution group of multiplicative functions. It is a consequence of the representation that the MFs can be defined in a natural way on the negative powers of the prime p which, in turn, leads to a natural extension of Schur-hook polynomials to negatively indexed Schur-hook polynomials. CONTENTS 0. Introduction 1. Ring of Isobaric Polynomials 2. The Companion Matrix and Core Polynomials 3 The Ring of Arithmetic Functions 4. Relation Between Arithmetic Functions and the WIP-Module 5. Examples 6. Specially Multiplicative Arithmetic Functions 7. Structure of the Convolution Group of Multiplicative Arithmetic Functions Reconsidered 8. Norms 0. INTRODUCTION In this paper we give a representation of the classical theory of multiplicative arithmetic functions (MF) in the ring of symmetric polynomials. The Dirichlet ring of arithmetic functions A∗ is well known to be a unique factorization domain.(see Cashwell and Everett [?]). Its ring theoretic properties have been investigated in, e.g., Rearick [?] and [?], Shapiro [?], Carroll and Gioia [?], MacHenry [?], MacHenry and Tudose [?]. The multiplicative arithmetic functions are units in this ring and their properties can be described locally, that is, at each prime number p, (see, e.g., McCarthy [?], Sivaramakishnan[?] and Vaidyanathswamy, [?]). It is this local behaviour which we take advantage of to construct a representation in terms of a certain class of symmetric polynomials

Abstract: In this article, we provide a comprehensive historical survey of different proofs of famous Euclid's theorem on the infinitude of prime numbers. The Bibliography of this article contains 99 references consisting of 24 textbooks and monographs, 73 articles (including 20 {\it Notes} published in {\it Amer. Math. Monthly} and a few unpublished works that are found on Internet Websites, especially on {\tt http:arxiv.org/}), one {\it Ph.D. thesis} and {\it Sloane's On-Line Encyclopedia of Integer Sequences}. The all references concerning to proofs of Euclid's theorem that use similar methods and ideas are exposed subsequently. Moreover, in Appendix we present a list of all 70 different proofs of Euclid's theorem presented here together with the corresponding reference(s), the name(s) of his (their) author(s) and the main method(s) and/or idea(s) used in it (them). This list is arranged by year of publication. In Section 2, we give a new simple proof of the {\it infinitude of primes}. The first step of our proof is based on Euclid's idea. The remaining of the proof is quite simple and elementary and it does not use the notion of divisibility.

Abstract: We develop the global theory of a strategy to tackle a program initiated by Bogomolov in 1990. That program aims at giving a group-theoretical recipe by which one can reconstruct function fields K | k with td(K | k) > 1 and k algebraically closed from the maximal pro-l abelian-by-central Galois group Π K c of K, where l is any prime number ≠char(k).

Abstract: We refine a result of W.P. Li and Wang (21) on the values of the form �1p1 + �2p 2 + �3p 2 + µ12 m 1 + ··· + µs2 m s , where p1,p2,p3 are prime numbers, m1,...,ms are positive integers, �1,�2,�3 are nonzero real numbers, not all of the same sign, �2/�3 is irrational andi/µi ∈ Q, for i ∈ {1,2,3}.

Abstract: The Riemann hypothesis states that all nontrivial zeros of the zeta function lie on the critical line $\Re(s)=1/2$. Hilbert and Polya suggested a possible approach to prove it, based on spectral theory. Within this context, some authors formulated the question: is there a quantum mechanical system related to the sequence of prime numbers? In this Letter we show that such a sequence is not zeta regularizable. Therefore, there are no physical systems described by self-adjoint operators with countably infinite number of degrees of freedom with spectra given by the sequence of primes numbers.

Abstract: In order for the simple theorem (that is primeval to the Pythagoras theorem) to be correct, mathematics as a whole, including by numbers, has to be 1:3 divergent, and the zero is to be -1. This manuscript in its primordial expression of the prime number 19 composite mocks the understanding of prime numbers/non-linear mathematics and numbers by current mathematics, western mathematics, and including its “suspect zero” which is in error. Mathematics is a theorem as created by the almighty and not a theory of a few so called greats in the history of mathematics, as there are absolute pointers in mathematics of which the number 19 is one. This is a vital Prime number and the very basis for the Pythagorean Theorem. It is obvious that the Vedic zero and gaps of 10, at 1:3 divergences is the correct zero, but the matter is too complex, and needs review of the prime number distribution as brought out in the published papers. We will in the course of our present publications with IJAMR and JAS, establish the mathematical fact of -1 as the correct new zero for all numbers...and a base offset of 0.5/60 or or the reciprocal of correct trigonometry and mathematical value of the mathematically correct π, which is very close to the established value.This will be published as part of the unified theorem.

Abstract: The notion of highly structured ring spectra of prime characteristic is made precise and is studied via the versal examples S//p for prime numbers p. These can be realized as Thom spectra, and therefore relate to other Thom spectra such as the unoriented bordism spectrum MO. We compute the Hochschild and Andr\'e-Quillen invariants of the S//p. Among other applications, we show that S//p is not a commutative algebra over the Eilenberg-Mac Lane spectrum HF_p, although the converse is clearly true, and that MO is not a polynomial algebra over S//2.

Abstract: Consider the Bianchi groups, namely the SL_2 groups over rings of imaginary quadratic integers. In the literature, there has been so far no example of p-torsion in the integral homology of the full Bianchi groups, for p a prime greater than the order of elements of finite order in the Bianchi group, which is at most 6. However, extending the scope of the computations, we can observe examples of torsion in the integral homology of the quotient space, at prime numbers as high as for instance p = 80737 at the discriminant -1747.

Abstract: We show that there is an unique connected quandle of order twice an odd prime number greater than 3. It has order 10 and is isomorphic to the conjugacy class of transpositions in the symmetric group of degree 5. This establishes a conjecture of L. Vendramin.

Abstract: We consider the second of Mullin's sequences of prime numbers related to Euclid's proof that there are infinitely many primes. We show in particular that it omits infinitely many primes, confirming a conjecture of Cox and van der Poorten.