Coprime Integers Research Articles

Products of three pairwise coprime integers in short intervals

AbstractThe existence of products of three pairwise coprime integers is investigated in short intervals of the form $(x,x+x^{\frac 12} ]$. A general theorem is proved which shows that such integer products exist provided there is a bound on the product of any two of them. A particular case of relevance to elliptic curve cryptography, where all three integers are of order $x^{\frac 13}$, is presented as a corollary to this result.

Geometric Exponents of Dilute Loop Models

The fractal dimensions of the hull, the external perimeter and of the red bonds are measured through Monte Carlo simulations for dilute minimal models, and compared with predictions from conformal field theory and SLE methods. The dilute models used are those first introduced by Nienhuis. Their loop fugacity is $\beta=-2 \cos(\pi/\bar{\kappa})$ where the parameter $\bar{\kappa}$ is linked to their description through conformal loop ensembles. It is also linked to conformal field theories through their central charges $c(\bar{\kappa})=13-6(\bar{\kappa}+\bar{\kappa}^{-1})$ and, for the minimal models of interest here, $\bar{\kappa}=p/p'$ where p and p′ are two coprime integers. The geometric exponents of the hull and external perimeter are studied for the pairs (p,p′)=(1,1),(2,3),(3,4),(4,5),(5,6),(5,7), and that of the red bonds for (p,p′)=(3,4). Monte Carlo upgrades are proposed for these models as well as several techniques to improve their speeds. The measured fractal dimensions are obtained by extrapolation on the lattice size H,V→∞. The extrapolating curves have large slopes; despite these, the measured dimensions coincide with theoretical predictions up to three or four digits. In some cases, the theoretical values lie slightly outside the confidence intervals; explanations of these small discrepancies are proposed.

Sequences of integers with missing quotients and dense points without neighbors

Let A be a pre-defined set of rational numbers. We say that a set of natural numbers S is an A-quotient-free set if no ratio of two elements in S belongs to A. We find the maximal asymptotic density and the maximal upper asymptotic density of A-quotient-free sets when A belongs to a particular class.It is known that in the case A={p,q}, where p, q are coprime integers greater than 1, the latter problem is reduced to the evaluation of the largest number of non-adjacent lattice points in a triangle whose legs lie on the coordinate axes. We prove that this number is achieved by choosing points of the same color in the checkerboard coloring.

Zeta-functions of harmonic theta-series and prime numbers

Speaking on rational prime numbers in various arithmetical sequences, it may be noted that no essential progress have been achieved for more than century and a half since famous Dirichlet theorem on prime numbers in arithmetic progressions (1837). Absolutely mystical is still the question on prime numbers in quadratic sequences, i.e., on prime numbers of the form an+bn+c, where a, b, c are rational coprime integers, and d = b − 4ac is not a rational square. The situation is not changing despite of considerable progress of the algebraic-analytical theory of integral quadratic forms reached after Dirichlet. Our purpose here is to draw the attention of numbertheorist to some analytical aspects of the theory of quadratic forms possibly related to the problem. In order to be more concrete, let us start from the the celebrated problem on prime numbers of the form 1+n. It is well known that the problem is closely related to reductions prime modules of certain elliptic curves with complex multiplications by Gauss integers a+ √ −1b, say, the curve y = x(x − 1). (1.1)

Remark on the completeness of an exponential type sequence

B. J. Birch [1] proved that all sufficiently large integers can be expressed as a sum of pairwise distinct terms of the form paqb, where p, q are given coprime integers greater than 1. Subsequently, Davenport pointed out that the exponent b can be bounded in terms of p and q. N. Hegyvari [3] gave an effective version of this bound. In this paper, we improve this bound by reducing one step.

Inhomogeneous approximation with coprime integers and lattice orbits

– Let (ξ, y) be a point in R 2 and ψ : N → R + a function. We investigate the problem of the existence of infinitely many pairs p, q of coprime integers such that |qξ + p − y| ≤ ψ(|q|). We give both unconditional results which are valid for every real pair (ξ, y) with ξ irrational, and metrical results valid for almost all points (ξ, y). We link the subject with density exponents of lattice orbits in R 2 .

A REFINED MODULAR APPROACH TO THE DIOPHANTINE EQUATION x2 + y2n = z3

Let n be a positive integer and consider the Diophantine equation of generalized Fermat type x2 + y2n = z3 in nonzero coprime integer unknowns x,y,z. Using methods of modular forms and Galois representations for approaching Diophantine equations, we show that for n ∈ {5,31} there are no solutions to this equation. Combining this with previously known results, this allows a complete description of all solutions to the Diophantine equation above for n ≤ 107. Finally, we show that there are also no solutions for n ≡ -1 (mod 6).

A note on the completeness of an exponential type sequence

For any given coprime integers p and q greater than 1, in 1959, B. J. Birch proved that all sufficiently large integers can be expressed as a sum of pairwise distinct terms of the form p a q b . As Davenport observed, Birch’s proof can be modified to show that the exponent b can be bounded in terms of p and q. In 2000, N. Hegyvari gave an effective version of this bound. The author improves this bound.

Slices for biparabolics of index 1

Let \( \mathfrak{a} \) be an algebraic Lie subalgebra of a simple Lie algebra \( \mathfrak{g} \) with index \( \mathfrak{a} \) ≤ rank \( \mathfrak{g} \). Let \( Y\left( \mathfrak{a} \right) \) denote the algebra of \( \mathfrak{a} \) invariant polynomial functions on \( {\mathfrak{a}^*} \). An algebraic slice for \( \mathfrak{a} \) is an affine subspace η + V with \( \eta \in {\mathfrak{a}^*} \) and \( V \subset {\mathfrak{a}^*} \) subspace of dimension index \( \mathfrak{a} \) such that restriction of function induces an isomorphism of \( Y\left( \mathfrak{a} \right) \) onto the algebra R[η + V] of regular functions on η + V. Slices have been obtained in a number of cases through the construction of an adapted pair (h, η) in which \( h \in \mathfrak{a} \) is ad-semisimple, η is a regular element of \( {\mathfrak{a}^*} \) which is an eigenvector for h of eigenvalue minus one and V is an h stable complement to \( \left( {{\text{ad}}\;\mathfrak{a}} \right)\eta \) in \( {\mathfrak{a}^*} \). The classical case is for \( \mathfrak{g} \) semisimple [16], [17]. Yet rather recently many other cases have been provided; for example, if \( \mathfrak{g} \) is of type A and \( \mathfrak{a} \) is a “truncated biparabolic” [12] or a centralizer [13]. In some of these cases (in particular when the biparabolic is a Borel subalgebra) it was found [13], [14], that η could be taken to be the restriction of a regular nilpotent element in \( \mathfrak{g} \). Moreover, this calculation suggested [13] how to construct slices outside type A when no adapted pair exists. This article makes a first step in taking these ideas further. Specifically, let \( \mathfrak{a} \) be a truncated biparabolic of index one. (This only arises if \( \mathfrak{g} \) is of type A and \( \mathfrak{a} \) is the derived algebra of a parabolic subalgebra whose Levi factor has just two blocks whose sizes are coprime.) In this case it is shown that the second member of an adapted pair (h, η) for \( \mathfrak{a} \) is the restriction of a particularly carefully chosen regular nilpotent element of \( \mathfrak{g} \). A by-product of our analysis is the construction of a map from the set of pairs of coprime integers to the set of all finite ordered sequences of ±1.

Generating New Commuting Coprime Matrix Pairs From Known Pairs

Commuting coprime integer matrices arise in signal processing in a number of contexts. This paper develops two ways, one nonlinear and the other linear, to generate new commuting coprime pairs from known pairs. The first method is based on computing powers of the initial pair of matrices, and the second method is based on appropriate types of linear combinations of the initial pair of matrices. This enriches the already known families of coprime matrices reported in recent literature. Several properties of the newly generated coprime pairs are also addressed.

Natural density of rectangular unimodular integer matrices

In this paper, we compute the natural density of the set of k×n integer matrices that can be extended to an invertible n×n matrix over the integers. As a corollary, we find the density of rectangular matrices with Hermite normal form Ok×(n-k)Ik. Connections with Cesàro’s Theorem on the density of coprime integers and Quillen–Suslin’s Theorem are also presented.

A minimal dynamical model for tidal synchronization and orbit circularization

We study tidal synchronization and orbit circularization in a minimal model that takes into account only the essential ingredients of tidal deformation and dissipation in the secondary body. In previous work we introduced the model (Escribano et al. in Phys. Rev. E, 78:036216, 2008); here we investigate in depth the complex dynamics that can arise from this simplest model of tidal synchronization and orbit circularization. We model an extended secondary body of mass m by two point masses of mass m/2 connected with a damped spring. This composite body moves in the gravitational field of a primary of mass M ≫ m located at the origin. In this simplest case oscillation and rotation of the secondary are assumed to take place in the plane of the Keplerian orbit. The gravitational interactions of both point masses with the primary are taken into account, but that between the point masses is neglected. We perform a Taylor expansion on the exact equations of motion to isolate and identify the different effects of tidal interactions. We compare both sets of equations and study the applicability of the approximations, in the presence of chaos. We introduce the resonance function as a resource to identify resonant states. The approximate equations of motion can account for both synchronization into the 1:1 spin-orbit resonance and the circularization of the orbit as the only true asymptotic attractors, together with the existence of relatively long-lived metastable orbits with the secondary in p:q (p and q being co-prime integers) synchronous rotation.

Mean value estimates of the error terms of Lehmer problem

Let p be an odd prime and a be an integer coprime with p. Denote by N(a, p) the number of pairs of integers b, c with bc ≡ a(mod p), 1 ≤ b, c < p and with b, c having different parity. The main purpose of this paper is to study the mean square value problem of (N(a, p) − 1/2 (p−1)) over interval (N, N + M] with M, N positive integers by using the analytic methods, and finally by obtaining a sharp asymptotic formula.

Bounding the Derived Length for a Given Set of Character Degrees

Let G be a finite solvable group with {1, a, b, c, ab, ac} as the character degree set, where a ,b, and c are pairwise coprime integers greater than 1. We show that the derived length of G is at most 4. This verifies that the Taketa inequality, dl(G) ≤ |cd(G)|, is valid for solvable groups with {1, a, b, c, ab, ac} as the character degree set. Also, as a corollary, we conclude that if a, b, c, and d are pairwise coprime integers greater than 1 and G is a solvable group such that cd(G) = {1, a, b, c, d, ac, ad, bc, bd}, then dl(G) ≤ 5. Finally, we construct a family of solvable groups whose derived lengths are 4 and character degree sets are in the form {1, p, b, pb, qp, pqp}, where p is a prime, q is a prime power of an odd prime, and b > 1 is integer such that p, q, and b are pairwise coprime. Hence, the bound 4 is the best bound for the derived length of solvable groups whose character degree set is in the form {1, a, b, c, ab, ac} for some pairwise coprime integers a, b, and c.

ON ELITE PRIMES OF PERIOD FOUR

A prime p is elite if all sufficiently large Fermat numbers Fn = 22n + 1 are quadratic nonresidues modulo p. In contrast, p is anti-elite if all sufficiently large Fn are quadratic residues modulo p. The sequence Fn modulo p is necessarily periodic. We give a sequence of pairwise coprime integers whose prime factors are each elite or anti-elite with period four.

On convergents formed from Diophantine equations

We compute upper and lower bounds for the approxi- mation of certain values ξ of hyperbolic and trigonometric functions by rationals x=y such that x; y satisfy Diophantine equations. We show that there are infinitely many coprime integers x; y such that |yξ − x| ≪ log log y log y and a Diophantine equation holds simultaneously relating x; y and some integer z. Conversely, all positive integers x; y with y ≥ c0 solving the Diophantine equation satisfy |yξ − x| ≫ log log y log y : Moreover, we approximate sin(πα) and cos(πα) by rationals in connection with solutions of a quadratic Diophantine equation when tan(πα=2) is a Liouville number.

Powers of Rational Numbers Modulo 1 Lying in Short Intervals

Let p > q > 1 be two coprime integers. We construct some positive numbers ξ such that the numbers ξ(p/q) n , n = 0, 1, 2, . . . , modulo 1 all lie in a short interval. Our results imply, for instance, that there exist three positive real numbers ξ, ζ, τ such that the inequalities ||ξ(5/3) n || 1/10 and ||τ (3/2)2n || < 14/45 hold for each integer \({n \geqslant 0}\).

Complementary results on the search of primes from lesser primes

In this short note we complete recently published results on the search of primes from lesser primes. The method is based on the concept of “compact” sets of primes and on the “fundamental property”, i.e., the fact that the decomposition in primes of the sum or the difference of two coprime integers never has common decomposition factors with these two coprime integers. The earlier finding is here enhanced, according to which a proportion of primes of about two thirds is to be found in the specified intervals where the result of the algorithm leads either to primes or the product of two primes.

New Einstein-Sasaki and Einstein spaces from Kerr-de Sitter

In this paper, which is an elaboration of our results in hep-th/0504225, we construct new Einstein-Sasaki spaces L^{p,q,r_1,...,r_{n-1}} in all odd dimensions D=2n+1\ge 5. They arise by taking certain BPS limits of the Euclideanised Kerr-de Sitter metrics. This yields local Einstein-Sasaki metrics of cohomogeneity n, with toric U(1)^{n+1} principal orbits, and n real non-trivial parameters. By studying the structure of the degenerate orbits we show that for appropriate choices of the parameters, characterised by the (n+1) coprime integers (p,q,r_1,...,r_{n-1}), the local metrics extend smoothly onto complete and non-singular compact Einstein-Sasaki manifolds L^{p,q,r_1,...,r_{n-1}}. We also construct new complete and non-singular compact Einstein spaces \Lambda^{p,q,r_1,...,r_n} in D=2n+1 that are not Sasakian, by choosing parameters appropriately in the Euclideanised Kerr-de Sitter metrics when no BPS limit is taken.

Newton functions generating symmetric fields and irreducibility of Schur polynomials

We prove (Theorem 1.1) that if e 0 > ⋯ > e r > 0 are coprime integers, then the Newton functions X 1 e i + ⋯ + X r e i , i = 0 , … , r , generate over Q the field of symmetric rational functions in X 1 , … , X r . This generalizes a previous result of us for r = 2 . This extension requires new methods, including: (i) a study of irreducibility and Galois-theoretic properties of Schur polynomials (Theorem 3.1), and (ii) the study of the dimension of the varieties obtained by intersecting Fermat hypersurfaces (Theorem 4.1). We shall also observe how these results have implications to the study of zeros of linear recurrences over function fields; in particular, we give (Theorem 4.2) a complete classification of the zeros of recurrences of order four with constant coefficients over a function field of dimension 1.