Abstract: This is a survey article on prime number races. Chebyshev noticed in the first half of the nineteenth century that for any given value of x, there always seem to be more primes of the form 4n+3 less than x then there are of the form 4n+1. Similar observations have been made with primes of the form 3n+2 and 3n+1, with primes of the form 10n+3/10n+7 and 10n+1/10n+9, and many others besides. More generally, one can consider primes of the form qn+a, qn+b, qn+c, >... for our favorite constants q, a, b, c, ... and try to figure out which forms are "preferred" over the others. In this paper, we describe these phenomena in greater detail and explain the efforts that have been made at understanding them.

Abstract: Contrary to what would be predicted on the basis of Cram\'er's model concerning the distribution of prime numbers, we develop evidence that the distribution of $\psi(x+H)- \psi(x)$, for $0\le x\le N$, is approximately normal with mean $\sim H$ and variance $\sim H\log N/H$, when $N^\delta \le H \le N^{1-\delta}$.

Abstract: The study of the distribution of prime numbers has fascinated mathematicians since antiquity. It is only in modern times, however, that a precise asymptotic law for the number of primes in arbitrarily long intervals has been obtained. For a real number x > 1, let π(x) denote the number of primes less than x. The prime number theorem is the assertion that $$ \mathop {\lim }\limits_{x \to \infty } \pi \left( x \right){\raise0.7ex\hbox{${}$} \!\mathord{\left/ {\vphantom {{} {}}}\right.\kern- ulldelimiterspace} \!\lower0.7ex\hbox{${}$}}\frac{x} {{\log (x)}} = 1. $$ This theorem was conjectured independently by Legendre and Gauss.

Abstract: It is well-known that Teichmuller discs that pass through "integer points'' of the moduli space of abelian differentials are very special: they are closed complex geodesics. However, the structure of these special Teichmuller discs is mostly unexplored: their number, genus, area, cusps, etc. We prove that in genus two all translation surfaces in H(2) tiled by a prime number n > 3 of squares fall into exactly two Teichmuller discs, only one of them with elliptic points, and that the genus of these discs has a cubic growth rate in n.

Abstract: We develop a network in which the natural numbers are the vertices. The decomposition of natural numbers by prime numbers is used to establish the connections. We perform data collapse and show that the degree distribution of these networks scales linearly with the number of vertices. We explore the families of vertices in connection with prime numbers decomposition. We compare the average distance of the network and the clustering coefficient with the distance and clustering coefficient of the corresponding random graph. In case we set connections among vertices each time the numbers share a common prime number the network has properties similar to a random graph. If the criterion for establishing links becomes more selective, only prime numbers greater than ${p}_{l}$ are used to establish links, where the network has high clustering coefficient.

Abstract: The order of every finite group G can be expressed as a product of coprime positive integers m1,...,mt such that the set of prime numbers divided mi is a connected component of the prime graph of G. The integers m1,...,mt are called the order components of G. Order components of a finite group are introduced in Chen (J. Algebra 15 (1996) 184). There exist some characterizations about alternating and symmetric groups. Some non-abelian simple groups are known to be uniquely determined by their order components. In this paper, we suppose that p=2a xb+1>5 be a prime number, where a,b>0 are positive integers and x>3 is an odd prime number. Then by using the classification of finite simple groups, we proved that Ap, Ap+1, Ap+2, Sp, Sp+1, are also uniquely determined by their order components. As corollaries of these results, the validity of a conjecture of J. G. Thompson and a conjecture of W. Shi and J. Bi both on An, where n=p, p+1 or p+2 are obtained. Also we generalize these conjectures for the groups Sn, where n=p, p+1.

Abstract: A Mersenne prime number is a prime number of the form 2k — 1. In this paper, we consider a Generalized Mersenne Prime (GMP) which is of the form R(k,p) = (p k -l)/(p - 1), where k,p and R(k,p) are prime numbers. For such a GMP, we then propose a much more efficient search algorithm for a special form of Multiple Recursive Generator (MRG) with the property of an extremely large period length and a high dimension of equidistribution. In particular, we find that (p k - l)/(p - 1) is a GMP, for k = 1511 and p = 2147427929. We then find a special form of MRG with order k = 1511 and modulus p = 2147427929 with the period length 1014100.5.Many other efficient and portable generators with various k ≤ 1511 are found and listed. Finally, for such a GMP and generator, we propose a simple and quick method of generating maximum period MRGs with the same order k. The readers are advised not to confuse GMP defined in this paper with other generalizations of the Mersenne Prime. For example, the term “Generalized Mersenne Number” (GMN) is used in Appendix 6.1 of FIPS-186-2, a publication by National Institute of Standards and Technology (NIST). In that document, GMN is a prime number that can be written as 2k ± 1 plus or minus a few terms of the form 2r.

Abstract: Schmutz Schaller's conjecture regarding the lengths of the hexagonal versus the lengths of the square lattice is shown to be true. The proof makes use of results from (computational) prime number theory.Using an identity due to Selberg, it is shown that, in principle, the conjecture can be resolved without using computational prime number theory. By our approach, however, this would require a huge amount of computation.

Abstract: These notes are based on my four lectures given at the Newton Institute in April 2004 during the Recent Perspectives in Random Matrix Theory and Number Theory Workshop. Their purpose is to introduce the reader to the analytic number theory necessary to understand Montgomery's work on the pair correlation of the zeros of the Riemann zeta-function and subsequent work on how this relates to prime numbers. A very brief introduction to Selberg's work on the moments of $S(T)$ is also given.

Abstract: In this paper we design a stream cipher that uses the algebraic structure of the multiplicative group $\bbbz_p^*$ (where p is a big prime number used in ElGamal algorithm), by defining a quasigroup of order $p-1$ and by doing quasigroup string transformations The cryptographical strength of the proposed stream cipher is based on the fact that breaking it would be at least as hard as solving systems of multivariate polynomial equations modulo big prime number $p$ which is NP-hard problem and there are no known fast randomized or deterministic algorithms for solving it Unlikely the speed of known ciphers that work in $\bbbz_p^*$ for big prime numbers $p$, the speed of this stream cipher both in encryption and decryption phase is comparable with the fastest symmetric-key stream ciphers

Abstract: Let a(n) denote the sum of the positive divisors of n. We say that n is perfect if a(n) = 2n. Currently there are no known odd perfect numbers. It is known that if an odd perfect number exists, then it must be of the form N = p α Π k j=1 q 2βj j , where p, q 1 ,..., q k are distinct primes and p ≡ α ≡ 1 (mod 4). Define the total number of prime factors of N as Ω(N):= a + 2Σ k j=1 β j . Sayers showed that Ω(N) > 29. This was later extended by Iannucci and Sorli to show that Ω(N) > 37. This paper extends these results to show that Ω(N) > 47.

Abstract: Under the assumption of the Riemann hypothesis the asymptotic value y/log x is known to hold for the number of primes in the short interval [x - y, x] for \(y = x^\alpha \) for every fixed \(\alpha < {1\over 2}\) We show under the assumption of the existence of exceptional Dirichlet characters the same asymptotic formula holds in the shorter intervals, for some \(\alpha < {1\over 2}\) \, in wide ranges of x depending on the characters

Abstract: We develop an algorithm for the construction of randomly shifted rank-1 lattice rules in d-dimensional weighted Sobolev spaces with a significantly reduced construction cost. The results shown here are an extension of earlier results by the present authors. In this new algorithm, the number of quadrature points n is a product of r distinct prime numbers p 1,…,p r. This allows us to reduce the construction cost to O(n(p 1 + … +p r)d 2), which represents a significant reduction, especially for large n. The constructed rules achieve a worst-case error bound with a rate of convergence of O(n(p 1 + δ p 2 -1/2 ... p r -1/2 ) for any δ > 0. Numerical experiments were carried out for r = 2, 3, 4 and 5. The results demonstrate that it can be advantageous to choose n as a product of up to 5 primes.

Abstract: kkt p be an odd prime number. In this article we study the distribution of p-class groups of cyclic number fields of degree p, and of cyclic extensions of degree p of an imaginary quadratic field whose class number is coprime to p. We formulate a heuristic principle predicting the distribution of the p-class groups as Galois modules, which is analogous to the Cohen-Lenstra heuristics concerning the prime-to-p-part of the class group, although in our case we have to fix the number of primes that ramify in the extensions considered. Using results of Gerth we are able to prove part of this conjecture. Furthermore, we present some numerical evidence for the conjecture.

Abstract: Counting rational points on Jacobian varieties of hyperelliptic curves over finite fields is very important for constructing hyperelliptic curve cryptosystems (HCC), but known algorithms for general curves over given large prime fields need very long running time. In this article, we propose an extremely fast point counting algorithm for hyperelliptic curves of type y 2 = x 5 + ax over given large prime fields Fp, e.g. 80-bit fields. For these curves, we also determine the necessary condition to be suitable for HCC, that is, to satisfy that the order of the Jacobian group is of the form l . c where l is a prime number greater than about 2 160 and c is a very small integer. We show some examples of suitable curves for HCC obtained by using our algorithm. We also treat curves of type y 2 = x 5 + a where a is not square in Fp.

Abstract: Extended digit multiplication can be an effective benchmark for comparing contemporary CPUs to other architectures and devices. The Great Internet Mersenne Prime Search (GIMPS), a distributed computing effort to find large prime numbers, has produced highly optimized code for multiplying large, multimillion digit numbers on Pentium processors. This paper presents a hardware large integer multiplier implemented in a Xilinx FPGA. The design, which utilizes an all-integer Fast Fourier Transform, is compared to a Pentium processor running the optimized GIMPS code. The results show that while a speed-up is achieved by the FPGA, the cost of the hardware likely outweighs any performance benefit.

Abstract: We study the fractal properties of the distances between consecutive primes. The distance sequence is found to be well described by a non-stationary exponential probability distribution. We propose an intensity-expansion method to treat this non-stationarity and we find that the statistics underlying the distance between consecutive primes is Gaussian and that, by transforming the distance sequence into a stationary one, the range of Gaussian randomness of the sequence increases.

Abstract: We prove that the number of quartic $S_4$--extensions of the rationals of given discriminant $d$ is $O_\eps(d^{1/2+\eps})$ for all $\eps>0$. For a prime number $p$ we derive that the dimension of the space of octahedral modular forms of weight 1 and conductor $p$ or $p^2$ is bounded above by $O(p^{1/2}\log(p)^2)$.

Abstract: Let G and G' be finite groups that have a common central p-subgroup Z for a prime number p, and let A and A' respectively be p-blocks of G/Z and G'/Z induced by p-blocks A and A' respectively of G and G', both of which have the same defect group. We prove that if A and A' are Morita equivalent via a certain special (A,A')-bimodule, then such a Morita equivalence lifts to a Morita equivalence between A and A'.

Abstract: Let E be an elliptic curve with complex multiplication by R, where R is an order of discriminant D < −4 of an imaginary quadratic field K. If a prime number p is decomposed completely in the ring class field associated with R, then E has good reduction at a prime ideal p of K dividing p and there exist positive integers u and υ such that 4p = u2 – Du;2. It is well known that the absolute value of the trace ap of the Frobenius endomorphism of the reduction of E modulo p is equal to u. We determine whether ap = u or ap = −u in the case where the class number of R is 2 or 3 and D is divisible by 3, 4 or 5.

Abstract: Let be a complete bipartite graph with two partite sets having and vertices, respectively. A -factorization of is a set of edge-disjoint -factors of which partition the set of edges of . When and is a prime number, Wang, in his paper "On -factorizations of a complete bipartite graph" (Discrete Math, 1994, 126: 359—364), investigated the -factorization of and gave a sufficient condition for such a factorization to exist. In the paper " -factorizations of complete bipartite graphs" (Discrete Math, 2002, 259: 301—306), Du and Wang extended Wang's result to the case that is any positive integer. In this paper, we give a sufficient condition for to have a -factorization. As a special case, it is shown that the Martin's BAC conjecture is true when for any positive integer .

Abstract: A Generalized Galois Ring (GGR) S is a finite nonassociative ring with identity of characteristic pn, for some prime number p, such that its top-factor is a semifield. It is well-known that if S is an associative Galois Ring (GR), then it contains a multiplicatively closed subset isomorphic to , the so-called Teichmuller Coordinate Set (TCS). In this paper we show that the existence of a TCS characterizes GR in the class of all GGR S such that the multiplicative loop is right (or left) primitive.