On integral Cayley sum graphs

Let S be a subset of a finite abelian group G. The Cayley sum graph Cay⁺(G, S) of G with respect to S is a graph whose vertex set is G and two vertices g and h are joined by an edge if and only if g + h ∈ S. We call a finite abelian group G a Cayley sum integral group if for every subset S of G, Cay⁺(G, S) is integral i.e., all eigenvalues of its adjacency matrix are integers. In this paper, we prove that all Cayley sum integral groups are represented by Z₃ and Zn₂ ⁿ, n ≥ 1, where Z_k is the group of integers modulo k. Also, we classify simple connected cubic integral Cayley sum graphs.     

Set-theoretic complete intersection monomial curves in $${\mathbb{P}^n}$$

In this paper, we give a sufficient numerical criterion for a monomial curve in a projective space to be a set-theoretic complete intersection. Our main result generalizes a similar statement proven by Keum for monomial curves in three-dimensional projective space. We also prove that there are infinitely many set-theoretic complete intersection monomial curves in the projective n?space for any suitably chosen n ? 1 integers. In particular, for any positive integers p, q, where gcd(p, q) = 1, the monomial curve defined by p, q, r is a set-theoretic complete intersection for every \({r \geq pq( q - 1)}\).     

Prime and composite integers close to powers of a number

We prove that, for any real numbers ξ ≠ 0 and ν, the sequence of integer parts [ξ2 ⁿ  + ν], n = 0, 1, 2, . . . , contains infinitely many composite numbers. Moreover, if the number ξ is irrational, then the above sequence contains infinitely many elements divisible by 2 or 3. The same holds for the sequence [ξ( ? 2) ⁿ  + ν _n ], n = 0, 1, 2, . . . , where ν ₀, ν ₁, ν ₂, . . . all lie in a half open real interval of length 1/3. For this, we show that if a sequence of integers x ₁, x ₂, x ₃, . . . satisfies the recurrence relation x _n+d  = cx _n  + F(x _n+1, . . . , x _n+d-1) for each n  ≥  1, where c ≠ 0 is an integer, \({F(z_1,\dots,z_{d-1}) \in \mathbb {Z}[z_1,\dots,z_{d-1}],}\) and lim _{n→ ∞}|x _n | = ∞, then the number |x _n | is composite for infinitely many positive integers n. The proofs involve techniques from number theory, linear algebra, combinatorics on words and some kind of symbolic computation modulo 3.     

On decomposability of finite groups

Let G be a finite group. A normal subgroup N of G is a union of several G-conjugacy classes, and it is called n-decomposable in G if it is a union of n distinct G-conjugacy classes. In this paper, we first classify finite non-perfect groups satisfying the condition that the numbers of conjugacy classes contained in its non-trivial normal subgroups are two consecutive positive integers, and we later prove that there is no non-perfect group such that the numbers of conjugacy classes contained in its non-trivial normal subgroups are 2, 3, 4 and 5.     

Non-Wieferich primes in number fields and <Emphasis Type="Italic">abc</Emphasis>-conjecture

Let K/Q be an algebraic number field of class number one and let O _K be its ring of integers. We show that there are infinitely many non-Wieferich primes with respect to certain units in O _K under the assumption of the abc-conjecture for number fields.     

A generalization of Williams’ results on Fourier coefficients of eta quotients

Recently, combining a product-to-sum formula and conditions for the non-representability of integers by certain ternary quadratic forms, Williams gave ten eta quotients such that their Fourier coefficients vanish for all positive integers n in each of infinitely many non-overlapping arithmetic progressions. In this paper, we generalize Williams’ results by utilizing theta function identities.     

Positive existential definability of multiplication from addition and the range of a polynomial

We consider the problem of recovering multiplication in the integers from enrichments of its additive structure, in the positive existential context. We prove that if a conjecture by Caporaso–Harris–Mazur holds, then for all integer-valued polynomials F of degree at least 2, multiplication is positive-existentially definable in (Z; 0, 1,+, R_F, =) where R_F is the unary relation F(Z). Similar results were only known for the polynomials F(t) = t² (under the Bombieri–Lang conjecture) and F(t) = tⁿ (under a generalization of the abc conjecture).     

Small ranks of central unit groups of integral group rings of alternating groups

We prove that the ranks of central unit groups of integral group rings of alternating groups of degrees greater than 38 are at least 11. The presented tables contain the ranks of all central unit groups of integral group rings of alternating groups of degrees at most 200. In particular, for every r ∈ {0, …, 10}, we obtain the complete list of integers n such that the central unit group of the integral group ring of the alternating group of degree n has rank r.     

Elements with prime and small indices in bicyclic biquadratic number fields

We give necessary and sufficient conditions for the existence of primitive algebraic integers with index A in totally complex bicyclic biquadratic number fields where A is an odd prime or a positive rational integer at most 10. We also determine all these elements and prove that there are infinitely many totally complex bicyclic biquadratic number fields containing elements with index A.     

Quotient and product sets of thin subsets of the positive integers

We study the cardinalities of A/A and AA for thin subsets A of the set of the first n positive integers. In particular, we consider the typical size of these quantities for random sets A of zero density and compare them with the sizes of A/A and AA for subsets of the shifted primes and the set of sums of two integral squares.     

On the existence of linear Pfaff systems with lower characteristic sets of positive Lebesgue <Emphasis Type="Italic">m</Emphasis>-measure

We prove the existence of an n-dimensional completely integrable Pfaff system with multidimensional time of dimension m ? 2, with bounded infinitely differentiable coefficients, and with the set of lower characteristic vectors of its solutions having positive Lebesgue m-measure.     

On realizability of sign patterns by real polynomials

The classical Descartes’ rule of signs limits the number of positive roots of a real polynomial in one variable by the number of sign changes in the sequence of its coefficients. One can ask the question which pairs of nonnegative integers (p, n), chosen in accordance with this rule and with some other natural conditions, can be the pairs of numbers of positive and negative roots of a real polynomial with prescribed signs of the coefficients. The paper solves this problem for degree 8 polynomials.     

Diophantine equations in separated variables

We study Diophantine equations of type \(f(x)=g(y)\), where both f and g have at least two distinct critical points (roots of the derivative) and equal critical values at at most two distinct critical points. Various classical families of polynomials \((f_n)_n\) are such that \(f_n\) satisfies these assumptions for all n. Our results cover and generalize several results in the literature on the finiteness of integral solutions to such equations. In doing so, we analyse the properties of the monodromy groups of such polynomials. We show that if f has coefficients in a field K of characteristic zero, and at least two distinct critical points and all distinct critical values, then the monodromy group of f is a doubly transitive permutation group. In particular, f cannot be represented as a composition of lower degree polynomials. Several authors have studied monodromy groups of polynomials with some similar properties. We further show that if f has at least two distinct critical points and equal critical values at at most two of them, and if \(f(x)=g(h(x))\) with \(g, h\in K[x]\) and \(\deg g>1\), then either \(\deg h\le 2\), or f is of special type. In the latter case, in particular, f has no three simple critical points, nor five distinct critical points.     

Arithmetically homogeneous functions: characterizations,stability and hyperstability

A function between two Abelian semigroups is arithmetically homogeneous of degree n if \( h(ix)=i^{n}h(x)\) for all positive integers i. We give new functional characterizations for arithmetically homogeneous functions, for polynomials of arithmetically homogeneous functions, and for Fréchet monomials. We also give large classes of control functions that provide generalized stability, or hyperstability for a homogeneous equation, and, consequently, new similar classes for classical monomial equations, so extending some known stability results.     

Are there <Emphasis Type="Italic">p</Emphasis>-adic knot invariants?

We suggest using the Hall–Littlewood version of the Rosso–Jones formula to define the germs of p-adic HOMFLY-PT polynomials for torus knots [m, n] as coefficients of superpolynomials in a q-expansion. In this form, they have at least the [m, n] ? [n, m] topological invariance. This opens a new possibility to interpret superpolynomials as p-adic deformations of HOMFLY polynomials and poses a question of generalizing to other knot families, which is a substantial problem for several branches of modern theory.     

Divisibility properties of hyperharmonic numbers

We extend Wolstenholme’s theorem to hyperharmonic numbers. Then, we obtain infinitely many congruence classes for hyperharmonic numbers using combinatorial methods. In particular, we show that the numerator of any hyperharmonic number in its reduced fractional form is odd. Then we give quantitative estimates for the number of pairs (n, r) lying in a rectangle where the corresponding hyperharmonic number \({ h_n^{(r)} }\) is divisible by a given prime number p. We also provide p-adic value lower bounds for certain hyperharmonic numbers. It is an open problem that given a prime number p, there are only finitely many harmonic numbers h _n which are divisible by p. We show that if we go to the higher levels r ≥  2, there are infinitely many hyperharmonic numbers \({ h_n^{(r)} }\) which are divisible by p. We also prove a finiteness result which is effective.     

A Munn type representation of abundant semigroups with a multiplicative ample transversal

In this note, we consider the Erd?s–Straus Diophantine equation

$$\begin{aligned} \frac{c}{n}=\frac{1}{x} + \frac{1}{y} + \frac{1}{z}, \end{aligned}$$

where n and c are positive integers with \(\gcd (n, c) = 1\). We provide a formula for the number f(n, c) of all positive integral solutions (x, y, z) of the equation. In 1948, Erd?s and Straus conjectured that \(f(n,4) \ge 1,\) for all integers \(n \ge 2\). Here, we solve the conjecture for a special case of n.

     

Types of Root Systems in Number Fields

We classify the types of root systems R in the rings of integers of number fields K such that the Weyl group W(R) lies in the group generated by Aut(K) and multiplications by the elements of K*.