Coprime Positive Integers Research Articles

Bounds for invariants of numerical semigroups and Wilf’s conjecture

Given coprime positive integers g_1< ldots < g_e, the Frobenius number F=F(g_1,ldots ,g_e) is the largest integer not representable as a linear combination of g_1,ldots ,g_e with non-negative integer coefficients. Let n denote the number of all representable non-negative integers less than F; Wilf conjectured that F+1 le e n. We provide bounds for g_1 and for the type of the numerical semigroup S=langle g_1,ldots ,g_e rangle in function of e and n, and use these bounds to prove that F+1 le q e n, where q= Bigg lceil frac{F+1}{g_1} Bigg rceil , and F+1 le e n^2. Finally, we give an alternative, simpler proof for the Wilf conjecture if the numerical semigroup S=langle g_1,ldots ,g_e rangle is almost-symmetric.

On the d-representation of integers

Let [Formula: see text] be two coprime positive integers. A positive integer [Formula: see text] is said to be [Formula: see text]-representable for [Formula: see text] if [Formula: see text] is the sum of terms taken from [Formula: see text] such that no one divides the other. Let [Formula: see text] denote the set of all positive integers that are [Formula: see text]-representable. For [Formula: see text], let [Formula: see text] be the maximum of the least terms of [Formula: see text]-representations of [Formula: see text] and let [Formula: see text] be the minimum of the largest terms of [Formula: see text]-representations of [Formula: see text]. In 1996, Erdős and Lewin conjectured that [Formula: see text] as [Formula: see text]. Recently, the authors of this paper confirmed this conjecture. This paper concerns the magnitudes of [Formula: see text] and [Formula: see text].

Coloured $$\mathfrak {sl}_r$$ invariants of torus knots and characters of $${\mathcal {W}}_r$$ algebras

Let $$p<p'$$ be a pair of coprime positive integers. In this note, generalizing Morton’s work in the case of $$\mathfrak {sl}_2$$ , we give a formula for the $$\mathfrak {sl}_r$$ Jones invariants of torus knots $$T (p,p')$$ coloured with the finite-dimensional irreducible representations $$L_r(n\Lambda _1)$$ . When $$r \le p$$ , we show that appropriate limits of the shifted (non-normalized, framing dependent) invariants calculated along $$L_r(nr\Lambda _1)$$ are essentially the characters of certain minimal model principal $${\mathcal {W}}$$ algebras of type $${{\,\textrm{A}\,}}$$ , namely, $${\mathcal {W}}_r(p,p')$$ , up to some factors independent of p and $$p'$$ but depending on r. In particular, these limits are essentially modular. We expect these limits to be the 0-tails of corresponding sequences of invariants. At the end, we formulate a conjecture on limits for $$p<r$$ .

Minimal Partitions with a Given s-Core and t-Core

Suppose s and t are coprime positive integers, and let sigma be an s-core partition and tau a t-core partition. In this paper, we consider the set {mathcal {P}}_{sigma ,tau }(n) of partitions of n with s-core sigma and t-core tau . We find the smallest n for which this set is non-empty, and show that for this value of n the partitions in {mathcal {P}}_{sigma ,tau }(n) (which we call (sigma ,tau )-minimal partitions) are in bijection with a certain class of (0, 1)-matrices with s rows and t columns. We then use these results in considering conjugate partitions: we determine exactly when the set {mathcal {P}}_{sigma ,tau }(n) consists of a conjugate pair of partitions, and when {mathcal {P}}_{sigma ,tau }(n) contains a unique self-conjugate partition.

Simple closed geodesics on regular tetrahedra in spherical space

We prove that there are finitely many simple closed geodesics on regular tetrahedra in spherical space. Also, for any pair of coprime positive integers , we find constants and depending on and and satisfying the inequality , such that a regular spherical tetrahedron with planar angle has a unique simple closed geodesic of type , up to tetrahedron isometry, whilst a regular spherical tetrahedron with planar angle has no such geodesic. Bibliography: 19 titles.

$${\pmb {{\mathbb {Z}}}}_p{\pmb {{\mathbb {Z}}}}_p[v]$$-additive cyclic codes are asymptotically good

We construct a class of $${\mathbb {Z}}_p{\mathbb {Z}}_p[v]$$ -additive cyclic codes, where p is a prime number and $$v^2=v$$ . We determine the asymptotic properties of the relative minimum distance and rate of this class of codes. We prove that, for any positive real number $$0<\delta <1$$ such that the p-ary entropy at $$\frac{k+l}{2}\delta $$ is less than $$\frac{1}{2}$$ , the relative minimum distance of the random code is convergent to $$\delta $$ and the rate of the random code is convergent to $$\frac{1}{k+l}$$ , where p, k, l are pairwise coprime positive integers.

Nontrivial upper bounds for the least common multiple of an arithmetic progression

In this paper, we establish some nontrivial and effective upper bounds for the least common multiple of consecutive terms of a finite arithmetic progression. Precisely, we prove that for any two coprime positive integers [Formula: see text] and [Formula: see text], with [Formula: see text], we have [Formula: see text] where [Formula: see text]. If in addition [Formula: see text] is a prime number and [Formula: see text], then we prove that for any [Formula: see text], we have [Formula: see text], where [Formula: see text]. Finally, we apply those inequalities to estimate the arithmetic function [Formula: see text] defined by [Formula: see text] ([Formula: see text]), as well as some values of the generalized Chebyshev function [Formula: see text].

ℤ4-Double Cyclic Codes Are Asymptotically Good

We construct a class of Z4-double cyclic codes generated by pairs of polynomials. Based on the probabilistic method, we prove the asymptotic properties of this class of codes: for any positive real number 0 <; δ <; 1 such that the 4-ary entropy at k+t/2 δ is less than 14, the rate of the random code is convergent to 1/k+t and the relative distance of the code is convergent to δ, where k and l are pairwise coprime positive odd integers. As a result, the Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> -double cyclic codes are asymptotically good.

The Diophantine equation (m2 + n2)x + (2mn)y = (m + n)2z

For fixed coprime positive integers [Formula: see text], [Formula: see text], [Formula: see text] with [Formula: see text] and [Formula: see text], there is a conjecture that the exponential Diophantine equation [Formula: see text] has only the positive integer solution [Formula: see text] for any positive integer [Formula: see text]. This is the analogue of Jésmanowicz conjecture. In this paper, we consider the equation [Formula: see text], where [Formula: see text] are coprime positive integers, and prove that the equation has no positive integer solution if [Formula: see text] and [Formula: see text].

Rigid connections on P1 via the Bruhat–Tits building

We apply the theory of fundamental strata of Bremer and Sage to find cohomologically rigid $G$-connections on the projective line, generalising the work of Frenkel and Gross. In this theory, one studies the leading term of a formal connection with respect to the Moy-Prasad filtration associated to a point in the Bruhat-Tits building. If the leading term is regular semisimple with centraliser a (not necessarily split) maximal torus $S$, then we have an $S$-toral connection. In this language, the irregular singularity of the Frenkel-Gross connection gives rise to the homogenous toral connection of minimal slope associated to the Coxeter torus $\mathcal{C}$. In the present paper, we consider connections on $\mathbb{G}_m$ which have an irregular homogeneous $\mathcal{C}$-toral singularity at zero of slope $i/h$, where $h$ is the Coxeter number and $i$ is a positive integer coprime to $h$, and a regular singularity at infinity with unipotent monodromy. Our main result is the characterisation of all such connections which are rigid.

Floer field theory for coprime rank and degree

We construct partial category-valued field theories in (2+1)-dimensions using Lagrangian Floer theory in moduli spaces of central-curvature unitary connections with fixed determinant of rank r and degree d where r,d are coprime positive integers. These theories associate to a closed, connected, oriented surface the Fukaya category of the moduli space, and to a connected bordism between two surfaces a functor between the Fukaya categories. We obtain the latter by combining Cerf theory with holomorphic quilt invariants.

The monochromatic column problem with a prime number of colors

Let p1,…,pn be a sequence of n pairwise coprime positive integers, P=p1⋯pn, and 0,…,m−1 be a sequence of m different colors. Let A be an n×mP matrix of colors in which row i consists of blocks of pi consecutive entries of the same color with colors 0 through m−1 repeated cyclically. The monochromatic column problem is to determine the number of columns of A in which every entry is the same color. The solution for a prime number of colors is provided.

Classification of irreducible weak modules over Virasoro vertex operator algebras

The classification of irreducible weak modules over the Virasoro vertex operator algebra [Formula: see text] is obtained in this paper. As one of the main results, we also classify all irreducible weak modules over the simple Virasoro vertex operator algebras [Formula: see text] for [Formula: see text] [Formula: see text], where [Formula: see text] are coprime positive integers.

The Hochschild cohomology rings of the numerical semigroup algebras of embedding dimension two

Let a and b be two coprime positive integers and k an arbitrary field. We determine the ring structure of the Hochschild cohomology of the numerical semigroup algebras k[sa,sb] of embedding dimension two (thus also complete intersections) in terms of generators and relations. In addition, we compute the Hilbert series of these cohomology rings.

The n-dimensional Stern–Brocot tree

This paper generalizes the Stern–Brocot tree to a tree that consists of all sequences of [Formula: see text] coprime positive integers. As for [Formula: see text] each sequence [Formula: see text] is the sum of a specific set of other coprime sequences, its Stern–Brocot set [Formula: see text], where [Formula: see text] is the degree of [Formula: see text] With an orthonormal base as the root, the tree defines a fast iterative structure on the set of distinct directions in [Formula: see text] and a multiresolution partition of [Formula: see text]. Basic proofs rely on a matrix representation of each coprime sequence, where the Stern–Brocot set forms the matrix columns. This induces a finitely generated submonoid [Formula: see text] of [Formula: see text], and a unimodular multidimensional continued fraction algorithm, also generalizing [Formula: see text]. It turns out that the [Formula: see text]-dimensional subtree starting with a sequence [Formula: see text] is isomorphic to the entire [Formula: see text]-dimensional tree. This allows basic combinatorial properties to be established. It turns out that also in this multidimensional version, Fibonacci-type sequences have maximal sequence sum in each generation.

Extension of results of Kostant and Han

Kostant proved that the mth coefficient of the $$(r^2-1)$$ th power of the Dedekind eta function is non-vanishing for $$r \ge \mathsf {max} \{4,m \}$$ and r an integer. This was generalized by Han to the real case applying the Nekrasov–Okounkov hook length formula. We extend these results to the complex case and identify coefficients of the zth power with values of polynomials $$P_m(-z)$$ . We assume that all non-trivial roots have a negative real part. Numerical data indicate that the quadratic bound can be replaced by a linear bound. Han showed by a reversion method that $$P_{m} \left( m +1\right) $$ is divisible by $$m +1$$ . We extend this result to $$P_{m} \left( \pm n \right) $$ , where m and n are two coprime positive integers. We obtain congruences modulo n in the case $$P_p\left( \pm n \right) $$ , where p is a prime. This leads to new non-vanishing results for the pth power of the Dedekind eta function, not covered in the theorems of Kostant and Han. We illustrate our results with numerical data and examples.

Rational noncrossing partitions for all coprime pairs

For coprime positive integers $a<b$, Armstrong, Rhoades, and Williams (2013) defined a set $NC(a,b)$ of rational noncrossing partitions, a subset of the ordinary noncrossing partitions of $\{1, \ldots, b-1\}$. Bodnar and Rhoades (2015) confirmed their conjecture that $NC(a,b)$ is closed under rotation and proved an instance of the cyclic sieving phenomenon for this rotation action. We give a definition of $NC(a,b)$ which works for all coprime $a$ and $b$ and prove closure under rotation and cyclic sieving in this more general setting. We also generalize noncrossing parking functions to all coprime $a$ and $b$, and provide a character formula for the action of $\mathfrak{S}_a \times \mathbb{Z}_{b-1}$ on $\mathsf{Park}^{NC}(a,b)$.

Enumeration of complementary-dual cyclic [formula omitted]-linear [formula omitted]-codes

Let Fq denote the finite field of order q,n be a positive integer coprime to q, and let t≥2 be an integer. In this paper, we provide enumeration formulae for all complementary-dual cyclic Fq-linear Fqt-codes of length n by placing ∗, ordinary and Hermitian trace inner products on Fqtn.

Modular Invariant Representations of the Superconformal Algebra

Abstract We compute the modular transformation formula of the characters for a certain family of (finitely or uncountably many) simple modules over the simple $\mathcal{N}=2$ vertex operator superalgebra of central charge $c_{p,p^{\prime }}=3\left (1-\frac{2p^{\prime }}{p}\right ),$ where (p, p′) is a pair of coprime positive integers such that p ≥ 2. When p′ = 1, the formula coincides with that of the $\mathcal{N}=2$ unitary minimal series found by F. Ravanini and S.-K. Yang. In addition, we study the properties of the corresponding “modular S-matrix”, which is no longer a matrix if p′≥ 2.

The Corners of Core Partitions

In this paper, we are concerned with counting corners of core partitions. We introduce the concepts of stitches and antistitches, which are pairs of cells in a quotient space which we call wrap-up space. We prove that the antistitches of a rational Dyck path are in bijection with the segments of structure sets of the corresponding core partition; therefore the corners of a core partition can be counted by the number of stitches or antistitches. Based on these results, for coprime positive integers $a$ and $b$, we give two essentially different formulae for the number of corners in all $(a,b)$-cores. This leads to an unexpected identity, expressing the rational Catalan numbers as weighed sums of binomial numbers. Moreover, we show that for an $(n,n+1)$-core partition $\lambda$ determined by a certain $(n,n+1)$-Dyck path $P$, the corners of $\lambda$ correspond to pairs of consecutive right steps in $P$. As a consequence, we show that the number of $(n,n+1)$-cores with $k$ corners is the Narayana number $N(n,k+1)$. We also extend these results to multicores.