Prime Integers Research Articles

On the distribution of norms of prime ideals of a given class in arithmetic progressions

Let C be a class of ideals of the ring of algebraic numbers of an imaginary quadratic field. Let l and q be relatively prime integers, \(1 \leqslant q \leqslant \log ^{A_1 } x\), and A1 τ; 1. An asymptotic formula for the number π1(x, q, l, C) of prime ideals belonging to the class C whose norms do not exceed x and lie in an arithmetic progression is obtained in this paper. Bibliography: 6 titles.

On the dynamic localization conditions for dc–ac electric fields proceeding beyond thenearest-neighbour description

Dynamic localization conditions proceeding beyond the nearest-neighbour description havebeen established by applying the quasi-energy description. The general energydispersion laws one deals with concern a periodic but commensurable dc–ac field likeE0+E1f(t), forwhich f(t) = f(t+T) and ωB = Pω/Q. Here and ω = 2π/T stand for the Bloch and ac field frequencies, respectively, whileP and Q are mutually prime integers. A reasonable centre of band generalization of such conditionshas been proposed.

On functions in p-adic BMO and the distribution of prime integers

We study the spaces BMO p of functions of bounded mean oscillation modeled on a p-adic martingale, and determine their relationship with the ordinary, continuous space BMO of functions of bounded mean oscillation. Somewhat surprisingly, these results are related to information about the distribution of primes.

155 弾性支持されたはりの分数調波振動

In this study, the occurrence of subharmonic tone in a nonlinear beam supported elastically subjected to a harmonic excitation is investigated. When the values of the spring constants in the springs supported at both ends have particular ones, a beam may have its natural frequencies in a ratio of prime integer. To analyze the problem in the above case, the purturbation method of multiple scales is used. The analysis shows that subharmonic tone of multimode type can occur. The occurrence of the subharmonic tone predicted theoretically is confirmed by numerical simulation.

The Growth of Selmer Ranks of an Abelian Variety with Complex Multiplication

Let K be a CM field and O be its ring of integers. Let p be an odd prime integer and p be a prime in K lying above p. Let F be a Galois extension of K unramified over p. For an Abelian variety A defined over F with complex multiplication by O, we study the variation of the p-ranks of the Selmer groups in pro-p algebraic extensions. We first study the Zpextension case. When K is quadratic imaginary and E is an elliptic curve, we also study the p-ranks of the Selmer groups in an unramified p-class field tower.

New Efficient MDS Array Codes for RAID Part II: Rabin-Like Codes for Tolerating Multiple (greater than or equal to 4) Disk Failures

For pt.1 see ibid., vol.54, no.9, p.1071-1080 (2005). A new class of binary maximum distance separable (MDS) array codes which are based on circular permutation matrices are introduced in this paper. These array codes are used for tolerating multiple (/spl ges/ 4) disk failures in redundant arrays of inexpensive disks (RAID) architecture. The size of the information part is m /spl times/ n, where n is the number of information disks and (m + 1) is a prime integer; the size of the parity-check part is m /spl times/ r, the minimum distance is r + 1, and the number of parity-check disks is r. In practical applications, m can be very large and n ranges from 20 to 50. The code rate is R = n/(n+r). These codes can be used for tolerating up to r disk failures, with very fast encoding and decoding. The complexities of encoding and decoding algorithms are O(rmn) and O(m/sup 3/r/sup 4/), respectively. When r = 4, there need to be 9mn XOR operations for encoding and (9n + 95)(m + 1) XOR operations for decoding.

The Modular Tree of Pythagoras

The Pythagorean triples that are relatively prime (called the primitive triples) have the elementary and beautiful characterization as integers x = m2 − n2, y = 2mn, z = m2 + n2 (when y is even) for relatively prime integers m and n of opposite parity. One can think of this as replacing the parameter t for the circle with the fraction m/n and then scaling. Our motivation for understanding the triples stems from the realization that one can enumerate the rational numbers on the line by using the modular group, in a sense reversing the Euclidean algorithm [2]. Now the line can be transformed by a linear fractional transformation to the circle. This transformation changes fractions to rational points on the circle, and after scaling this process gives rise to Pythagorean triples. Roughly speaking, we can establish a correspondence of a Pythagorean triple [m2 − n2, 2mn,m2 + n2] in which m and n are relatively prime with a matrix belonging to SL2(Z) (the group of two-by-two integral matrices of determinant one) whose entries depend on m and n. Since the modular group = PSL2(Z) = SL2(Z)/{±I } is essentially a free group, it follows that there is an underlying tree structure to Pythagorean triples. Making this tree structure and its connection to the modular group explicit is a bit delicate, but the payoff is worth the effort. Our main results can be summarized as follows:

THE INTEGRALITY OF DIHEDRAL QUANDLE COCYCLE INVARIANTS

We calculate the dihedral quandle cocycle invariants of twist-spins of alternating odd pretzel knots. The calculation leads us to the conclusion that there exist non-ribbon 2-knots which admit a non-trivial coloring by the dihedral quandle Rp and all of whose cocycle invariants derived from ℤp-valued 3-cocycles on Rp take value in ℤ ⊂ ℤ[ℤp] for any odd prime integer p.

The abc–conjecture for Algebraic Numbers

The abc–conjecture for the ring of integers states that, for every e > 0 and every triple of relatively prime nonzero integers (a, b, c) satisfying a + b = c, we have max(|a|, |b|, |c|) ≤ rad(abc)1 + e with a finite number of exceptions. Here the radical rad(m) is the product of all distinct prime factors of m.

Canonical [formula omitted]-dimension of algebraic groups

We describe a way to compute the p -relative version of the Berhuy–Reichstein canonical dimension for an arbitrary split semisimple algebraic group over an arbitrary field of an arbitrary characteristic ( p is any prime integer). The canonical p -dimension is computed for all split simple groups of classical types.

New Efficient MDS Array Codes for RAID Part I: Reed-Solomon-Like Codes for Tolerating Three Disk Failures

This paper presents a class of binary maximum distance separable (MDS) array codes for tolerating disk failures in redundant arrays of inexpensive disks (RAID) architecture based on circular permutation matrices. The size of the information part is m/spl times/n, the size of the parity-check part is m/spl times/3, and the minimum distance is 4, where n is the number of information disks, the number of parity-check disks is 3, and (m+1) is a prime integer. In practical applications, m can be very large and n is from 20 to 50. The code rate is R=n/(n+3). These codes can be used for tolerating three disk failures. The encoding and decoding of the Reed-Solomon-like codes are very fast. There need to be 3mn XOR operations for encoding and (3mn+9(m+1)) XOR operations for decoding.

The RegularChains library in MAPLE

Performing calculations modulo a set of relations is a basic technique in algebra. For instance, computing the inverse of an integer modulo a prime integer or computing the inverse of the complex number 3 + 2<i>t</i> modulo the relation ℓ<sup>2</sup> + 1 = 0. Computing modulo a set <i>S</i> containing more than one relation requires from <i>S</i> to have some mathematical structure. For instance, computing the inverse of <i>p</i> = <i>x</i> + <i>y</i> modulo <i>S</i> = {<i>x</i><sup>2</sup> + <i>y</i> + 1,<i>y</i><sup>2</sup> + <i>x</i> + 1} is difficult unless one realizes that this question is equivalent to computing the inverse of <i>p</i> modulo <i>C</i> = {<i>x</i><sup>4</sup> + 2<i>x</i><sup>2</sup> + <i>x</i> + 2,<i>y</i> + <i>x</i><sup>2</sup> + 1}. Indeed, from there one can simplify <i>p</i> using <i>y</i> = -<i>x</i><sup>2</sup> - 1 leading to <i>q</i> = -<i>x</i><sup>2</sup> + <i>x</i> - 1 and compute the inverse of <i>q</i> modulo <i>x</i><sup>4</sup> + 2<i>x</i><sup>2</sup> + <i>x</i> + 2 (using the extended Euclidean algorithm) leading to -1/2<i>x</i><sup>3</sup> - 1/2<i>x</i>. One commonly used mathematical structure for a set of algebraic relations is that of a <i>Gröbner basis.</i> It is particularly well suited for deciding whether a quantity is null or not modulo a set of relations. For inverse computations, the notion of a <i>regular chain</i> is more adequate. For instance, computing the inverse of <i>p</i> = <i>x</i> + <i>y</i> modulo the set <i>C</i> = {<i>y</i><sup>2</sup> - 2<i>x</i> + 1,<i>x</i><sup>2</sup> - 3<i>x</i> + 2}, which is both a Gröbner basis and a regular chain, is easily answered in this latter point of view. Indeed, it naturally leads to consider the GCD of <i>p</i> and <i>C<inf>y</inf></i> = <i>y</i><sup>2</sup> - 2<i>x</i> + 1 modulo the relation <i>C<inf>x</inf></i> = <i>x</i><sup>2</sup> - 3<i>x</i> + 2 = 0, which is [EQUATION] This shows that <i>p</i> has no inverse if <i>x</i> = 1 and has an inverse (which can be computed and which is -<i>y</i> + 2) if <i>x</i> = 2.

Stable Manifolds and Homoclinic Points Near Resonances in the Restricted Three-Body Problem

The restricted three-body problem describes the motion of a massless particle under the influence of two primaries of masses $1-\mu$ and $\mu$ that circle each other with period equal to $2\pi$. For small $\mu$, a resonant periodic motion of the massless particle in the rotating frame can be described by relatively prime integers $p$ and $q$, if its period around the heavier primary is approximately $2\pi p/q$, and by its approximate eccentricity $e$. We give a method for the formal development of the stable and unstable manifolds associated with these resonant motions. We prove the validity of this formal development and the existence of homoclinic points in the resonant region. In the study of the Kirkwood gaps in the asteroid belt, the separatrices of the averaged equations of the restricted three-body problem are commonly used to derive analytical approximations to the boundaries of the resonances. We use the unaveraged equations to find values of asteroid eccentricity below which these approximations will not hold for the Kirkwood gaps with $q/p$ equal to 2/1, 7/3, 5/2, 3/1, and 4/1. Another application is to the existence of asymmetric librations in the exterior resonances. We give values of asteroid eccentricity below which asymmetric librations will not exist for the 1/7, 1/6, 1/5, 1/4, 1/3, and 1/2 resonances for any $\mu$ however small. But if the eccentricity exceeds these thresholds, asymmetric librations will exist for $\mu$ small enough in the unaveraged restricted three-body problem.

Hamiltonian Minimal Lagrangian Cones in ${\mathbb C}^{m}$

We give a correspondence among a Hamiltonian minimal Lagrangian cone in ${\mathbb C}^{m}$, a Legendrian minimal Legendrian submanifold in the unit sphere $S^{2m-1}(1)$ and a Hamiltonian minimal Lagrangian submanifold in the complex projective space ${\mathbb C}P^{m-1}$. As an application of this result, we prove that a Hamiltonian minimal Lagrangian cone in ${\mathbb C}^{m}$ such that the first Betti number of its link is 0 must be a special Lagrangian cone. Moreover, we construct Hamiltonian minimal (non-minimal) Lagrangian cones in ${\mathbb C}^{3}$ with a toroidal link, which are parametrized by a triple of relatively prime positive integers $(p,q,r)$, and discuss their Hamiltonian stabilities.

Putting competing orders in their place near the Mott transition

We describe the localization transition of superfluids on two-dimensional lattices into commensurate Mott insulators with average particle density p/q (p, q relatively prime integers) per lattice site. For bosons on the square lattice, we argue that the superfluid has at least q degenerate species of vortices which transform under a projective representation of the square lattice space group (a PSG). The formation of a single vortex condensate produces the Mott insulator, which is required by the PSG to have density wave order at wavelengths of q/n lattice sites (n integer) along the principle axes; such a second-order transition is forbidden in the Landau-Ginzburg-Wilson framework. We also discuss the superfluid-insulator transition in the direct boson representation, and find that an interpretation of the quantum criticality in terms of deconfined fractionalized bosons is only permitted at special values of q for which a permutative representation of the PSG exists. We argue (and demonstrate in detail in a companion paper: L. Balents et al., cond-mat/0409470) that our results apply essentially unchanged to electronic systems with short-range pairing, with the PSG determined by the particle density of Cooper pairs. We also describe the effect of static impurities in the superfluid: the impurities locally break the degeneracy between the q vortex species, and this induces density wave order near each vortex. We suggest that such a theory offers an appealing rationale for the local density of states modulations observed by Hoffman et al. (cond-mat/0201348) in STM studies of the vortex lattice of BSCCO, and allows a unified description of the nucleation of density wave order in zero and finite magnetic fields. We note signatures of our theory that may be tested by future STM experiments.

B-free numbers in short arithmetic progressions

Given a sequence B of relatively prime positive integers with the sum of inverses finite, we investigate the problem of finding B-free numbers in short arithmetic progressions.

Arithmetical properties of wendt's determinant

Wendt's determinant of order n is the circulant determinant W n whose ( i , j ) -th entry is the binomial coefficient n | i - j | , for 1 ⩽ i , j ⩽ n , where n is a positive integer. We establish some congruence relations satisfied by these rational integers. Thus, if p is a prime number and k a positive integer, then W p k ≡ 1 ( mod p k ) and W np k ≡ W n ( mod p ) . If q is another prime, distinct from p, and h any positive integer, then W p h q k ≡ W p h W q k ( mod pq ) . Furthermore, if p is odd, then W p ≡ 1 + p 2 p - 1 p - 1 - 1 ( mod p 5 ) . In particular, if p ⩾ 5 , then W p ≡ 1 ( mod p 4 ) . Also, if m and n are relatively prime positive integers, then W m W n divides W mn .

On labeling the vertices of products of complete graphs with distance constraints

AbstractVariations of Hale's channel assignment problem, the L(j, k)‐labeling problem and the radio labeling problem require the assignment of integers to the vertices of a graph G subject to various distance constraints. The λj,k‐number of G and the radio number of G are respectively the minimum span among all L(j, k)‐labelings, and the minimum span plus 1 of all radio labelings of G (defined in the Introduction). In this paper, we establish the λj,k‐number of ∏ K for pairwise relatively prime integers t1 < t2 < … < tq, t1 ≥ 2. We also show the existence of an infinite class of graphs G with radio number |V(G)| for any diameter d(G). © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2005

On the appearance of primes in linear recursive sequences

We present an application of difference equations to number theory by considering the set of linear second-order recursive relations, , U0 = 0, U1 = 1, and , , where R and Q are relatively prime integers and n ∈ {0,1,...}. These equations describe the set of extended Lucas sequences, or rather, the Lehmer sequences. We add that the rank of apparition of an odd prime p in a specific Lehmer sequence is the index of the first term that contains p as a divisor. In this paper, we obtain results that pertain to the rank of apparition of primes of the form 2np ± 1. Upon doing so, we will also establish rank of apparition results under more explicit hypotheses for some notable special cases of the Lehmer sequences. Presently, there does not exist a closed formula that will produce the rank of apparition of an arbitrary prime in any of the aforementioned sequences.

Lower estimates for the number of closed trajectories of generalized billiards

The mathematical study of periodic billiard trajectories is a classical question that goes back to George Birkhoff. A billiard is the motion of a particle in the absence of field of force. Trajectories of such a particle are geodesics. A billiard ball rebounds from the boundary of a given domain making the angle of incidence equal the angle of reflection. Let k be a fixed integer. Birkhoff proved a lower estimate for the number of closed billiard trajectories of length k in an arbitrary plane domain. We give a general definition of a closed billiard trajectory when the billiard ball rebounds from a submanifold of a Euclidean space. Besides, we show how in this case one can apply the Morse inequalities using the natural symmetry (a closed polygon may be considered starting at any of its vertices and with the reversed direction). Finally, we prove the following estimate. Let M be a smooth closed m-dimensional submanifold of a Euclidean space, and let p > 2 be a prime integer. Then M has at least $$\frac{{(B - 1)((B - 1)^{p - 1} - 1)}}{{2p}} + \frac{{mB}}{2}(p - 1)$$ closed billiard trajectories of length p. Bibliography: 7 titles.