Abstract: The aim of this paper is to discuss variants for elliptic curves of some deep conjectures of classical cyclotomic Iwasawa theory. Let F be a finite extension of Q, p an odd prime number, and F cyc the cyclotomic Zp-extension of Q. Let K(F cyc) denote the maximal unramified abelian p-extension of F cyc, in which every prime of F cyc above p splits completely, and define Y (F cyc) to be the Galois group of K(F cyc) over F cyc. Put = G(F cyc/F ), and write ( ) for the Iwasawa algebra of . Iwasawa proved that Y (F cyc) is always a finitely generated torsion ( )-module, and he conjectured [12] that in fact Y (F cyc) is always a finitely generated Zp-module. At present, this conjecture has only been proven when F is abelian over Q (see [5],[26]). Perhaps surprisingly, it does not seem to have been pointed out in the literature that there is a precise analogue of Iwasawa’s conjecture for elliptic curves over F cyc. Let E be an elliptic curve over F , and let

Abstract: The RSA public-key cryptosystem is an algorithm that converts input data to an unrecognizable encryption and converts the unrecognizable data back into its original decryption form. The security of the RSA public-key cryptosystem is based on the difficulty of factoring the product of two large prime numbers. This paper demonstrates to factor the product of two large prime numbers, and is a breakthrough in basic biological operations using a molecular computer. In order to achieve this, we propose three DNA-based algorithms for parallel subtractor, parallel comparator, and parallel modular arithmetic that formally verify our designed molecular solutions for factoring the product of two large prime numbers. Furthermore, this work indicates that the cryptosystems using public-key are perhaps insecure and also presents clear evidence of the ability of molecular computing to perform complicated mathematical operations.

Abstract: Acknowledgments. Author's Note. Introduction. Entries A to Z. abc conjecture. abundant number. AKS algorithm for primality testing. aliquot sequences (sociable chains). almost-primes. amicable numbers. amicable curiosities. Andrica's conjecture. arithmetic progressions, of primes. Aurifeuillian factorization. average prime. Bang's theorem. Bateman's conjecture. Beal's conjecture, and prize. Benford's law. Bernoulli numbers. Bernoulli number curiosities. Bertrand's postulate. Bonse's inequality. Brier numbers. Brocard's conjecture. Brun's constant. Buss's function. Carmichael numbers. Catalan's conjecture. Catalan's Mersenne conjecture. Champernowne's constant. champion numbers. Chinese remainder theorem. cicadas and prime periods. circle, prime. circular prime. Clay prizes, the. compositorial. concatenation of primes. conjectures. consecutive integer sequence. consecutive numbers. consecutive primes, sums of. Conway's prime-producing machine. cousin primes. Cullen primes. Cunningham project. Cunningham chains. decimals, recurring (periodic). the period of 1/13. cyclic numbers. Artin's conjecture. the repunit connection. magic squares. deficient number. deletable and truncatable primes. Demlo numbers. descriptive primes. Dickson's conjecture. digit properties. Diophantus (c. AD 200 d. 284). Dirichlet's theorem and primes in arithmetic series. primes in polynomials. distributed computing. divisibility tests. divisors (factors). how many divisors? how big is d(n)? record number of divisors. curiosities of d(n). divisors and congruences. the sum of divisors function. the size of sigma(n). a recursive formula. divisors and partitions. curiosities of sigma(n). prime factors. divisor curiosities. economical numbers. Electronic Frontier Foundation. elliptic curve primality proving. emirp. Eratosthenes of Cyrene, the sieve of. Erdos, Paul (1913-1996). his collaborators and Erdos numbers. errors. Euclid (c. 330-270 BC). unique factorization. &Radic 2 is irrational. Euclid and the infinity of primes. consecutive composite numbers. primes of the form 4n +3. a recursive sequence. Euclid and the first perfect number. Euclidean algorithm. Euler, Leonhard (1707-1783). Euler's convenient numbers. the Basel problem. Euler's constant. Euler and the reciprocals of the primes. Euler's totient (phi) function. Carmichael's totient function conjecture. curiosities of phi(n). Euler's quadratic. the Lucky Numbers of Euler. factorial. factors of factorials. factorial primes. factorial sums. factorials, double, triple ... factorization, methods of. factors of particular forms. Fermat's algorithm. Legendre's method. congruences and factorization. how difficult is it to factor large numbers? quantum computation. Feit-Thompson conjecture. Fermat, Pierre de (1607-1665). Fermat's Little Theorem. Fermat quotient. Fermat and primes of the form x 2 + y 2 . Fermat's conjecture, Fermat numbers, and Fermat primes. Fermat factorization, from F 5 to F 30 . Generalized Fermat numbers. Fermat's Last Theorem. the first case of Fermat's Last Theorem. Wall-Sun-Sun primes. Fermat-Catalan equation and conjecture. Fibonacci numbers. divisibility properties. Fibonacci curiosities. Edouard Lucas and the Fibonacci numbers. Fibonacci composite sequences. formulae for primes. Fortunate numbers and Fortune's conjecture. gaps between primes and composite runs. Gauss, Johann Carl Friedrich (1777-1855). Gauss and the distribution of primes. Gaussian primes. Gauss's circle problem. Gilbreath's conjecture. GIMPS-Great Internet Mersenne Prime Search. Giuga's conjecture. Giuga numbers. Goldbach's conjecture. good primes. Grimm's problem. Hardy, G. H. (1877-1947). Hardy-Littlewood conjectures. heuristic reasoning. a heuristic argument by George Polya. Hilbert's 23 problems. home prime. hypothesis H. illegal prime. inconsummate number. induction. jumping champion. k-tuples conjecture, prime. knots, prime and composite. Landau, Edmund (1877-1938). left-truncatable prime. Legendre, A. M. (1752-1833). Lehmer, Derrick Norman (1867-1938). Lehmer, Derrick Henry (1905-1991). Linnik's constant. Liouville, Joseph (1809-1882). Littlewood's theorem. the prime numbers race. Lucas, Edouard (1842-1891). the Lucas sequence. primality testing. Lucas's game of calculation. the Lucas-Lehmer test. lucky numbers. the number of lucky numbers and primes. "random" primes. magic squares. Matijasevic and Hilbert's 10th problem. Mersenne numbers and Mersenne primes. Mersenne numbers. hunting for Mersenne primes. the coming of electronic computers. Mersenne prime conjectures. the New Mersenne conjecture. how many Mersenne primes? Eberhart's conjecture. factors of Mersenne numbers. Lucas-Lehmer test for Mersenne primes. Mertens constant. Mertens theorem. Mills' theorem. Wright's theorem. mixed bag. multiplication, fast. Niven numbers. odd numbers as p + 2a 2 . Opperman's conjecture. palindromic primes. pandigital primes. Pascal's triangle and the binomial coefficients. Pascal's triangle and Sierpinski's gasket. Pascal triangle curiosities. patents on prime numbers. Pepin's test for Fermat numbers. perfect numbers. odd perfect numbers. perfect, multiply. permutable primes. pi, primes in the decimal expansion of. Pocklington's theorem. Polignac's conjectures. Polignac or obstinate numbers. powerful numbers. primality testing. probabilistic methods. prime number graph. prime number theorem and the prime counting function. history. elementary proof. record calculations. estimating p(n). calculating p(n). a curiosity. prime pretender. primitive prime factor. primitive roots. Artin's conjecture. a curiosity. primordial. primorial primes. Proth's theorem. pseudoperfect numbers. pseudoprimes. bases and pseudoprimes. pseudoprimes, strong. public key encryption. pyramid, prime. Pythagorean triangles, prime. quadratic residues. residual curiosities. polynomial congruences. quadratic reciprocity, law of. Euler's criterion. Ramanujan, Srinivasa (1887-1920). highly composite numbers. randomness, of primes. Von Sternach and a prime random walk. record primes. some records. repunits, prime. Rhonda numbers. Riemann hypothesis. the Farey sequence and the Riemann hypothesis. the Riemann hypothesis and sigma(n), the sum of divisors function. squarefree and blue and red numbers. the Mertens conjecture. Riemann hypothesis curiosities. Riesel number. right-truncatable prime. RSA algorithm. Martin Gardner's challenge. RSA Factoring Challenge, the New. Ruth-Aaron numbers. Scherk's conjecture. semi-primes. sexy primes. Shank's conjecture. Siamese primes. Sierpinski numbers. Sierpinski strings. Sierpinski's quadratic. Sierpinski's phi(n) conjecture. Sloane's On-Line Encyclopedia of Integer Sequences. Smith numbers. Smith brothers. smooth numbers. Sophie Germain primes. safe primes. squarefree numbers. Stern prime. strong law of small numbers. triangular numbers. trivia. twin primes. twin curiosities. Ulam spiral. unitary divisors. unitary perfect. untouchable numbers. weird numbers. Wieferich primes. Wilson's theorem. twin primes. Wilson primes. Wolstenholme's numbers, and theorems. more factors of Wolstenholme numbers. Woodall primes. zeta mysteries: the quantum connection. Appendix A: The First 500 Primes. Appendix B: Arithmetic Functions. Glossary. Bibliography. Index.

Abstract: A famous theorem of Szemeredi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy between structure and randomness, which in turn leads (roughly speaking) to a decomposition of any object into a structured (low-complexity) component and a random (discorrelated) component. Important examples of these types of decompositions include the Furstenberg structure theorem and the Szemeredi regularity lemma. One recent application of this dichotomy is the result of Green and Tao establishing that the prime numbers contain arbitrarily long arithmetic progressions (despite having density zero in the integers). The power of this dichotomy is evidenced by the fact that the Green-Tao theorem requires surprisingly little technology from analytic number theory, relying instead almost exclusively on manifestations of this dichotomy such as Szemeredi's theorem. In this paper we survey various manifestations of this dichotomy in combinatorics, harmonic analysis, ergodic theory, and number theory. As we hope to emphasize here, the underlying themes in these arguments are remarkably similar even though the contexts are radically different.

Abstract: A long standing and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an amazing fusion of methods from analytic number theory and ergodic theory, Ben Green and Terence Tao showed that for any positive integer k, there exist infinitely many arithmetic progressions of length k consisting only of prime numbers. This is an introduction to some of the ideas in the proof, concentrating on the connections to ergodic theory.

Abstract: We consider some problems of analytic number theory for elliptic curves which can be considered as analogues of classical questions around the distribution of primes in arithmetic progressions to large moduli, and of the question of twin primes This leads to some local results on the distribution of the group structures of elliptic curves defined over a prime finite field, exhibiting an interesting dichotomy for the occurence of the possible groups (Note : This paper was initially written in 2000/01, but after a four year wait for a referee report, it is now withdrawn and deposited in the arXiv)

Abstract: Recently, Goubin, Mauduit, Rivat and Sarkozy have given three constructions for large families of binary sequences. In each of these constructions the sequence is defined by modulo ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> p$ congruences where $p$ is a prime number. In this paper the three constructions are extended to the case when the modulus is of the form $pq$ where $p$, $q$ are two distinct primes not far apart (note that the well-known Blum-Blum-Shub and RSA constructions for pseudorandom sequences are also of this type). It is shown that these modulo $pq$ constructions also have certain strong pseudorandom properties but, e.g., the (``long range'') correlation of order $4$ is large (similar phenomenon may occur in other modulo $pq$ constructions as well).

Abstract: We use short divisor sums to approximate prime tuples and moments for primes in short intervals By connecting these results to classical moment problems we are able to prove that a positive proportion of consecutive primes are within a quarter of the average spacing between primes

Abstract: According to Goldbach conjecture, any even number can be broken up as the sum of two prime numbers: n = p + q . We construct a network where each node is a prime number and corresponding to every even number n, we put a link between the component primes p and q. In most cases, an even number can be broken up in many ways, and then we chose one decomposition with a probability | p - q | α . Through computation of average shortest distance and clustering coefficient, we conclude that for α > - 1.8 the network is of small world type and for α - 1.8 it is of regular type. We also present a theoretical justification for such behaviour.

Abstract: We remove logarithmic factors in error term estimates in asymptotic formulas for the number of solutions of a class of additive congruences modulo a prime number.

Abstract: Let p be a prime number, which, for simplicity, we shall always assume odd. In the Iwasawa theory of an elliptic curve E over a number field k one has to distinguish between curves which do or do not admit complex multiplication (CM). For CMelliptic curves their deep arithmetic properties and the link between their Selmer group and special values of their Hasse-Weil L-functions are not only described by the (one-variable) main conjecture corresponding to the cyclotomic Zp-extension kcyc of k, but also by the (two-variable) main conjecture corresponding to the extension k∞ = k(Ep∞) which arises by adjoining the p-power division points Ep∞ of E. Moreover, both conjectures are proven by Rubin [36] in the case that k is imaginary quadratic and E has CM by the ring of integers Ok of k. Also for non-CM elliptic curves one would like to at least formulate a main conjecture over the trivialzing extension k∞, but for lack of both an algebraic as well as analytic p-adic L-function this has not been achieved. The aim of this paper is to establish, under certain conditions, the existence of an algebraic p-adic L-function, viz as an element of the first K-group K1(ΛT ) ∼= ΛT /[Λ × T ,Λ × T ] of a localization ΛT of the usual Iwasawa algebra Λ = Λ(G) of the Galois group G = G(k∞/k). Here, for a ring R, we denote by R× its group of units. By the Weil-pairing, kcyc is contained in k∞ and we put H = G(k∞/kcyc) and Γ = G(kcyc/k). Furthermore we write m(H) for the kernel of the canonical surjective ring homomorphism

Abstract: 1. INTRODUCTION. Two of the most ubiquitous objects in mathematics are the sequence of prime numbers and the binomial coefficients (and thus Pascal’s triangle). A connection between the two is given by a well-known characterization of the prime numbers: Consider the entries in the kth row of Pascal’s triangle, without the initial and final entries. They are all divisible by k if and only if k is a prime. It is the purpose of this article to present a triangular array of numbers similar to Pascal’s triangle and to prove a corresponding criterion for the twin prime pairs .A further goal is to place all this in the context of some classical orthogonal polynomials and to relate it to some recent work of John D’Angelo. To begin, and for the sake of completeness, we present a short proof of the Pascal triangle criterion. First suppose that k = p is prime. Then we see that in p j = p! j! ( p − j)! (1 ≤ j ≤ p − 1)

Abstract: In this paper we prove that the alternating groupsAn, forn=p, p+1, p+2 and symmetric groupsSn, forn=p, p+1, wherep>=3 is a prime number, can be uniquely determined by their order components. As one of the important consequence of this characterization we show that the simple groupsAn, wheren=p, p+1, p+2 andp>=3 is prime, satisfy in Thompson's conjecture and Shi's conjecture.

Abstract: The multiplication by a constant (say, by 2) acts on the set Z/nZ of residues (mod n) as a dynamical system, whose cycles relatively prime to n all have a common period T(n) and whose orbits consist each of elements, forming a geometrical progression or residues. The paper provides many new facts on the arithmetical properties of these periods and orbits (generalizing the Fermat's small theorem, extended by Euler to the case where n is not a prime number). The chaoticity of the orbit is measured by some randomness parameter, comparing the distances distribution of neighbouring points of the orbit with a similar distribution for T randomly chosen residues (which is binominal). The calculations show some kind of repulsion of neighbours, avoiding to be close to other members of the same orbit. A similar repulsion is also observed for the prime numbers, providing their distributions nonrandomness, and for the arithmetical progressions of the residues, whose nonrandomness degree is similar to that of the primes. The paper contains also many conjectures, including that of the infinity of the pairs of prime numbers of the form (q, 2q+1), like (3,7),(11,23) ,(23,47)on one side and that on the structure of some ideals in the multiplicative semigroup of odd integers – on the other.

Abstract: We establish a new bound for the exponential sum \begin{eqnarray*} \sum_{x\in\mathcal{X}}\Big|\sum_{y\in \mathcal{Y}}\gamma(y)\exp(2\pi i a \lambda^{xy}/p)\Big|, \end{eqnarray*} where $\lambda$ is an element of the residue ring modulo a large prime number $p,$ $\mathcal{X}$ and $\mathcal{Y}$ are arbitrary subsets of the residue ring modulo $p-1$ and $\gamma(n)$ are any complex numbers with $|\gamma(n)| \le 1.$ In particular, we improve several previously known bounds.

Abstract: We show that for infinitely many prime numbers p there are at least \log\log p / \log\log\log p distinct residue classes modulo p that are not congruent to n! for any integer n.

Abstract: We give an example of a vector bundle e on a relative curve C → Spec ℤ such that the restriction to the generic fiber in characteristic zero is semistable but such that the restriction to positive characteristic p is not strongly semistable for infinitely many prime numbers p Moreover, under the hypothesis that there exist infinitely many Sophie Germain primes, there are also examples such that the density of primes with nonstrongly semistable reduction is arbitrarily close to one

Abstract: Diffie-Hellman (DH) is a well-known cryptographic algorithm used for secure key exchange. The first appearance of DH was in 1976. The algorithm allows two users to exchange a symmetric secret key through an insecure wired or wireless channel and without any prior secrets. DH works under the domain of integers Z*/sub n/ where n = p. Here, p and /spl alpha/ are the two parameters of DH where p is a large prime number and /spl alpha/ is a generator selected from the cyclic group Z*/sub n/. In this paper, we propose two modifications of DH. The first modification is to change the domain to integer with n=2p/sup t/ where Z*/sub n/ is still cyclic and the second modification is to change the domain to Gaussian arithmetic Z*/sub n/. After implementing the three algorithms we found that the symmetric key size derived from the two modified algorithms is much greater than the classical one. Moreover, attacking the two modified algorithms using Pohlig-Hellman algorithm, using the same prime value p and private value a or b, needs much more time than the classical one.

Abstract: Let G be a finite p-group, where p is a prime number, and a ∈ G. Denote by Cl(a) = {gag−1| g ∈ G} the conjugacy class of a in G. Assume that |Cl(a)| = p n . Then Cl(a) Cl(a−1) = {xy | x ∈ Cl(a), y ∈ Cl(a−1)} is the union of at least n(p − 1) + 1 distinct conjugacy classes of G.

Abstract: “God created numbers, all the rest has been created by Man. . . ”. With greatest esteem to Leopold Kronecker, one of the founders of the contemporary theory of numbers, it is impossible to agree with him in both the divine origin of number and Man’s creation of mathematics. I propound herein the idea that numbers, their relations, and all mathematics in general are objective realities of our world. A part of science is not only understanding things, but also studying the relations that are objective realities in nature. In this work I am going to consider a problem concerning the distribution of rational numbers along the number line and also in the number plane, and the relation of this distribution to resonance phenomena and stability of oscillating systems in low linear perturbations. Any oscillating process involving at least two interacting oscillators is necessarily linked to abstract numbers — ratios between the oscillation periods. This fact displays a close relationship between such sections of science as the physical theory of oscillations and the abstract theory of numbers. As is well known, the rational numbers are distributed on the number line everywhere compactly, so this problem statement that a function of their distribution exists might be thought false, as the case of prime numbers. But, as we will see below, it is not false — a rational numbers distribution function has an objective reality, manifest in numerous physical phenomena of Nature. This thesis will become clearer if we consider the “number lattice” introduced by Minkowski (Fig. 1). Therein are given all points of coordinates p and q which are related to numerators and denominators, respectively. If we exclude all points of the Minkowski lattice with coordinates have a common divisor different from unity, this plane will contain only “rational points” p/q (the non-cancelled fractions). Their distribution in the plane is defined by a sequence of numbers forming a rational series (Fig. 1). This simplest drawing shows that rational numbers are distributed inhomogeneously in the Minkowski number plane. It is easy to see that this distribution is symmetric with respect to the axis p=q. Numbers of columns (and rows) in intervals, limited by this axis and one of the coordinate axes, are equal to Euler functions — the numbers less than m and relatively prime with m. Therefore, if we expand the number lattice infinitely, the average density of rational numbers in the plane (the ratio between the number of rational numbers and the

Abstract: A. Fermat proved that every (positive) prime number p of the type p = 4k + 1 can be represented as a sum of two squares of integers. This easily implies that a positive integer n can be written as a sum of two squares of integers if and only if all prime factors of n of the form 4k + 3 have even exponents in the standard decomposition of n. B. Lagrange showed that every positive integer is the sum of four squares of inte-

Abstract: Let S be a non-empty finite set of prime numbers and, for each p in S , let Z p denote the ring of p -adic integers. Let F be an abelian extension over the rational field such that the Galois group of F over some subfield of F with finite degree is topologically isomorphic to the additive group of the direct product of Z p for all p in S . We shall prove that each of certain arithmetic progressions contains only finitely many prime numbers l for which the l -class group of F is nontrivial. This result implies our conjecture in [3] that the set of prime numbers l for which the l -class group of F is trivial has natural density 1 in the set of all prime numbers.

Abstract: The original Gelfond-Schnirelman method, proposed in 1936, uses polynomials with in- teger coefficients and small norms on (0,1) to give a Chebyshev-type lower bound in prime number theory. We study a generalizationof this method for polynomials in many variables. Ourmain result is a lower bound for the integral of Chebyshev's -function, expressed in terms of the weighted capacity. This extends previous work of Nair and Chudnovsky, and connects the subject to the potential theory with external fields generated by polynomial-type weights. We also solve the corresponding potential theoretic problem, by finding the extremal measure and its support.

Abstract: We show that the two cuspidal unipotent characters of a finite Chevalley group 7( ) have Schur index 2, provided that is an even power of a (sufficiently large) prime number such that 1 mod 4. The proof uses a refinement of Kawanaka’s generalized Gelfand‐Graev representations and some explicit computations with the CHEVIE computer algebra system.