Nontrivial Divisor Research Articles

A Multiplicative Property for Zero-Sums II

For $n\geq 1$, let $C_n$ denote a cyclic group of order $n$. Let $G=C_n\oplus C_{mn}$ with $n\geq 2$ and $m\geq 1$, and let $k\in [0,n-1]$. It is known that any sequence of $mn+n-1+k$ terms from $G$ must contain a nontrivial zero-sum of length at most $mn+n-1-k$. The associated inverse question is to characterize those sequences with maximal length $mn+n-2+k$ that fail to contain a nontrivial zero-sum subsequence of length at most $mn+n-1-k$. For $k\leq 1$, this is the inverse question for the Davenport Constant. For $k=n-1$, this is the inverse question for the $\eta(G)$ invariant concerning short zero-sum subsequences. For $C_n\oplus C_n$ and $k\in [2,n-2]$, with $n\geq 5$ prime, it was conjectured in a paper of Grynkiewicz, Wang and Zhao that they must have the form $S=e_1^{[n-1]}\boldsymbol{\cdot} e_2^{[n-1]}\boldsymbol{\cdot} (e_1+e_2)^{[k]}$ for some basis $(e_1,e_2)$, with the conjecture established in many cases and later extended to composite moduli $n$. In this paper, we focus on the case $m\geq 2$. Assuming the conjectured structure holds for $k\in [2,n-2]$ in $C_n\oplus C_n$, we characterize the structure of all sequences of maximal length $mn+n-2+k$ in $C_n\oplus C_{mn}$ that fail to contain a nontrivial zero-sum of length at most $mn+n-1-k$, showing they must either have the form $S=e_1^{[n-1]}\boldsymbol{\cdot} e_2^{[sn-1]}\boldsymbol{\cdot} (e_1+e_2)^{[(m-s)n+k]}$ for some $s\in [1,m]$ and basis $(e_1,e_2)$ with $\mathsf{ord}(e_2)=mn$, or else have the form $S=g_1^{[n-1]}\boldsymbol{\cdot} g_2^{[n-1]}\boldsymbol{\cdot} (g_1+g_2)^{[(m-1)n+k]}$ for some generating set $\{g_1,g_2\}$ with $\mathsf{ord}(g_1+g_2)=mn$. In view of prior work, this reduces the structural characterization for a general rank two abelian group to the case $C_p\oplus C_p$ with $p$ prime. Additionally, we give a new proof of the precise structure in the case $k=n-1$ for $m=1$. Combined with known results, our results unconditionally establish the structure of extremal sequences in $G=C_n\oplus C_{mn}$ in many cases, including when $n$ is only divisible by primes at most $7$, when $n\geq 2$ is a prime power and $k\leq \frac{2n+1}{3}$, or when $n$ is composite and $k=n-d-1$ or $n-2d+1$ for a proper, nontrivial divisor $d\mid n$.

A multiplicative property for zero-sums I

Let G=Cn×Cn, where Cn denotes a cyclic group of order n, and let k∈[0,n−1]. We study the structure of sequences of terms from G with maximal length |S|=2n−2+k that fail to contain a nontrivial zero-sum subsequence of length at most 2n−1−k. For k≤1, this is the inverse question for the Davenport Constant. For k=n−1, this is the inverse question for the η(G) invariant concerning short zero-sum subsequences. The structure in both these cases (known respectively as Property B and Property C) was established in a two-step process: first verifying the multiplicative property that, if the structural description holds when n=n1 and n=n2, then it holds when n=n1n2, and then resolving the case n prime separately. When n is prime, the structural characterization for k∈[2,2n+13] was recently established, showing S must have the form S=e1[n−1]⋅e2[n−1]⋅(e1+e2)[k] for some basis (e1,e2) for G. It was conjectured that this also holds for k∈[2,n−2] (when n is prime). In this paper, we extend this conjecture by dropping the restriction that n be prime and establish the following multiplicative result. Suppose k=kmn+kn with km∈[0,m−1] and kn∈[0,n−1]. If the conjectured structure holds for km in Cm×Cm and for kn in Cn×Cn, then it holds for k in Cmn×Cmn. This reduces the full characterization question for n and k to the prime case. Combined with known results, this unconditionally establishes the structure for extremal sequences in G=Cn×Cn in many cases, including when n is only divisible by primes at most 7, when n≥2 is a prime power and k≤2n+13, or when n is composite and k=n−d−1 or n−2d+1 for a proper, nontrivial divisor d|n.

On the least common multiple of binary linear recurrence sequences

We present a method for estimating the least common multiple of a large class of binary linear recurrence sequences. Let P, Q, R0 and R1 be fixed integers, and let R=(Rn)n≥0 be the recurrence sequence defined by Rn+2=PRn+1−QRn for all n≥0. Under some conditions on the parameters, we determine a rational nontrivial divisor for Lk,n:= lcm(Rk,Rk+1,…,Rn), for all positive integers n and k, such that n≥k. As consequences, we derive nontrivial effective lower bounds for Lk,n, and we establish an asymptotic formula for log(Ln,n+m), where m is a fixed positive integer. Denoting by (Fn)n the usual Fibonacci sequence, we prove, for example, that for any m≥1, we have loglcm(Fn,Fn+1,…,Fn+m)∼n(m+1)logΦ( as n→+∞), where Φ denotes the golden ratio. We conclude the paper with some interesting identities and properties regarding the least common multiple of Lucas sequences.

Upper Bounds for Cyclotomic Numbers

Let q be a power of a prime p, let k be a nontrivial divisor of q-1 and write e=(q-1)/k. We study upper bounds for cyclotomic numbers (a,b) of order e over the finite field 𝔽 q . A general result of our study is that (a,b)≤3 for all a,b∈ℤ if p>(14) k/ord k (p) . More conclusive results will be obtained through separate investigation of the five types of cyclotomic numbers: (0,0),(0,a),(a,0),(a,a) and (a,b), where a≠b and a,b∈{1,...,e-1}. The main idea we use is to transform equations over 𝔽 q into equations over the field of complex numbers on which we have more information. A major tool for the improvements we obtain over known results is new upper bounds on the norm of cyclotomic integers.

On Some Formulas for Kaprekar Constants

Let b ≥ 2 and n ≥ 2 be integers. For a b-adic n-digit integer x, let A (resp. B) be the b-adic n-digit integer obtained by rearranging the numbers of all digits of x in descending (resp. ascending) order. Then, we define the Kaprekar transformation T ( b , n ) ( x ) : = A - B . If T ( b , n ) ( x ) = x , then x is called a b-adicn-digit Kaprekar constant. Moreover, we say that a b-adic n-digit Kaprekar constant x is regular when the numbers of all digits of x are distinct. In this article, we obtain some formulas for regular and non-regular Kaprekar constants, respectively. As an application of these formulas, we then see that for any integer b ≥ 2 , the number of b-adic odd-digit regular Kaprekar constants is greater than or equal to the number of all non-trivial divisors of b. Kaprekar constants have the symmetric property that they are fixed points for recursive number theoretical functions T ( b , n ) .

A complete characterization of pretty good state transfer on paths

We give a complete characterization of pretty good state transfer on paths between any pair of vertices with respect to the quantum walk model determined by the XY-Hamiltonian. If n is the length of the path, and the vertices are indexed by the positive integers from 1 to n, with adjacent vertices having consecutive indices, then the necessary and sufficient conditions for pretty good state transfer between vertices a and b are that (a) a + b = n + 1, (b) n + 1 has at most one positive odd non-trivial divisor, and (c) if n = 2^t r - 1, for r odd and r \neq 1, then a is a multiple of 2^{t - 1}.

Number theoretic properties of the commutative ring Zn

This paper deals with the number theoretic properties of non-unit elements of the ring Zn. Let D be the set of all non-trivial divisors of a positive integer n. Let D1 and D2 be the subsets of D having the least common multiple which are incongruent to zero modulo n with every other element of D and congruent to zero modulo n with at least one another element of D, respectively. Then D can be written as the disjoint union of D1 and D2 in Zn. We explore the results on these sets based on all the characterizations of n. We obtain a formula for enumerating the cardinality of the set of all non-unit elements in Zn whose principal ideals are equal. Further, we present an algorithm for enumerating these sets of all non-unit elements.

Some properties and applications of non-trivial divisor functions

The jth divisor function d_j, which counts the ordered factorisations of a positive integer into j positive integer factors, is a very well-known multiplicative arithmetic function. However, the non-multiplicative jth non-trivial divisor functionc_j, which counts the ordered factorisations of a positive integer into j factors each of which is greater than or equal to 2, is rather less well studied. Additionally, we consider the associated divisor functionc_j^{(r)}, for rge 0, whose definition is motivated by the sum-over divisors recurrence for d_j. We give an overview of properties of d_j, c_j and c_j^{(r)}, specifically regarding their Dirichlet series and generating functions as well as representations in terms of binomial coefficient sums and hypergeometric series. Noting general inequalities between the three types of divisor function, we then observe how their ratios can be expressed as binomial coefficient sums and hypergeometric series, and find explicit Dirichlet series and Euler products for some of these. As an illustrative application of the non-trivial and associated divisor functions, we show how they can be used to count principal reversible square matrices of the type considered by Ollerenshaw and Brée and so sum-and-distance systems of integers.

Five-vertex common divisor character degree graphs of non-solvable groups

For a finite group G, the common divisor graph attached to G is a character degree graph whose vertices are the degrees of the non-linear irreducible complex characters of G, and distinct vertices a, b are adjacent when they have a non-trivial common divisor. In this article, we determine all graphs with five vertices that may occur as for a non-solvable group G.

Algebra of Symmetries of Three-Frequency Resonance: Reduction of a Reducible Case to an Irreducible Case

For the three-frequency quantum resonance oscillator, the reducible case, where the frequencies are integer and at least one pair of frequencies has a nontrivial common divisor, is studied. It is shown how the description of the algebra of symmetries of such an oscillator can be reduced to the irreducible case of pairwise coprime integer frequencies. Polynomial algebraic relations are written, and irreducible representations and coherent states are constructed.

A certain family of subgroups of ℤ𝑛⋆ is weakly pseudo-free under the general integer factoring intractability assumption

AbstractLet {\mathbb{G}_{n}} be the subgroup of elements of odd order in the group {\mathbb{Z}^{\star}_{n}}, and let {\mathcal{U}(\mathbb{G}_{n})} be the uniform probability distribution on {\mathbb{G}_{n}}. In this paper, we establish a probabilistic polynomial-time reduction from finding a nontrivial divisor of a composite number n to finding a nontrivial relation between l elements chosen independently and uniformly at random from {\mathbb{G}_{n}}, where {l\geq 1} is given in unary as a part of the input. Assume that finding a nontrivial divisor of a random number in some set N of composite numbers (for a given security parameter) is a computationally hard problem. Then, using the above-mentioned reduction, we prove that the family {((\mathbb{G}_{n},\mathcal{U}(\mathbb{G}_{n}))\mid n\in N)} of computational abelian groups is weakly pseudo-free. The disadvantage of this result is that the probability ensemble {(\mathcal{U}(\mathbb{G}_{n})\mid n\in N)} is not polynomial-time samplable. To overcome this disadvantage, we construct a polynomial-time computable function {\nu\colon D\to N} (where {D\subseteq\{0,1\}^{*}}) and a polynomial-time samplable probability ensemble {(\mathcal{G}_{d}\mid d\in D)} (where {\mathcal{G}_{d}} is a distribution on {\mathbb{G}_{\nu(d)}} for each {d\in D}) such that the family {((\mathbb{G}_{\nu(d)},\mathcal{G}_{d})\mid d\in D)} of computational abelian groups is weakly pseudo-free.

Almost Primes in Thin Orbits of Pythagorean Triangles

Abstract Let $F=x^2+y^2-z^2$, and let $x_0 \in \mathbb{Z}^3$ be a $primitive$ solution to $F(x_0)=0$, e.g., so that its coordinates share no nontrivial divisor. Let $\Gamma \leq \mathrm{SO_F(\mathbb{Z})}$ be a thin subgroup. We consider the resulting thin orbits of Pythagorean triples $x_0 \cdot \Gamma$—specifically which hypotenuses, areas, and products of all three coordinates arise. We produce infinitely many $R$-almost primes in these three cases whenever $\Gamma$ has exponent $\delta_\Gamma>\delta_0(R)$ for explicit $R$, $\delta_0$.

Mode control for microring resonators with inner-wall gratings

Microring resonators with angular gratings distributed along the inner wall are proposed to achieve single-mode microlasers. Based on a numerical investigation using the finite element method, we find that the inner-wall gratings create a periodic perturbation and couple the whispering-gallery modes into low-angular-momentum components, resulting in an enhancement of radiation loss. By properly arranging the inner-wall grating protrusion number as a nontrivial divisor of 2ν, we can have the mode with azimuthal mode number ν as the only high Q mode for single-mode operation. The impacts of grating depth, microring width, and grating duty cycle are investigated by 2D simulation for the optimization of mode coupling and radiation loss. Finally, the mode selection mechanism is verified based on a 3D simulation.

Spaces of surface group representations

Let $$\Gamma _g$$ denote the fundamental group of a closed surface of genus $$g \ge 2$$ . We show that every geometric representation of $$\Gamma _g$$ into the group of orientation-preserving homeomorphisms of the circle is rigid, meaning that its deformations form a single semi-conjugacy class. As a consequence, we give a new lower bound on the number of topological components of the space of representations of $$\Gamma _g$$ into $${{\mathrm{Homeo}}}_+(S^1)$$ . Precisely, for each nontrivial divisor $$k$$ of $$2g-2$$ , there are at least $$|k|^{2g} + 1$$ components containing representations with Euler number $$\frac{2g-2}{k}$$ . Our methods apply to representations of surface groups into finite covers of $${{\mathrm{PSL}}}(2,\mathbb {R})$$ and into $${{\mathrm{Diff}}}_+(S^1)$$ as well, in which case we recover theorems of W. Goldman and J. Bowden. The key technique is an investigation of stability phenomena for rotation numbers of products of circle homeomorphisms using techniques of Calegari–Walker. This is a new approach to studying deformation classes of group actions on the circle, and may be of independent interest.

Computing prime divisors in an interval

We address the problem of finding a nontrivial divisor of a composite integer when it has a prime divisor in an interval. We show that this problem can be solved in time of the square root of the interval length with a similar amount of storage, by presenting two algorithms; one is probabilistic and the other is its derandomized version.

Permutation polynomials and their differential properties over residue class rings

This paper mainly focuses on permutation polynomials over the residue class ring ZN, where N>3 is composite. We have proved that for the polynomial f(x)=a1x+a2x2+⋯+akxk with integral coefficients, f(x)modN permutes ZN if and only if f(x)modN permutes Sμ for all μ∣N, where Sμ={0<t<N:gcd(N,t)=μ} for μ≠N and SN={0}. Based on it, we give a lower bound of the differential uniformities for such permutation polynomials, that is, δ(f)≥N#Sa, where a is the biggest nontrivial divisor of N. Especially, f(x) cannot be APN permutations over the residue class ring ZN. It is also proved that f(x)modN and (f(x)+x)modN cannot permute ZN at the same time when N is even.

On the distribution of some integers related to perfect and amicable numbers

Let s′(n) = ∑ d|n, 1<d<n d be the sum of the nontrivial divisors of the natural number n, where nontrivial excludes both 1 and n. For example, s′(20) = 2 + 4 + 5 + 10 = 21. A natural number n is called quasiperfect if s′(n) = n, while n and m are said to form a quasiamicable pair if s′(n) = m and s′(m) = n; in the latter case, both n and m are called quasiamicable numbers. In this paper, we prove two statistical theorems about these classes of numbers. First, we show that the count of quasiperfect n ≤ x is at most x 1 4, as x → ∞. In fact, we show that for each fixed a, there are at most x 1 4+o(1) natural numbers n ≤ x with σ(n) ≡ a (mod n) and σ(n) odd. (Quasiperfect n satisfy these conditions with a = 1.) For fixed δ 6= 0, define the arithmetic function sδ(n) := σ(n)−n− δ. Thus, s1 = s′. Our second theorem says that the number of n ≤ x which are amicable with respect to sδ is at most x/(log x) .

The strength of sharply bounded induction requires [formula omitted

We show that the arithmetical theory T 2 0 + Σ ˆ 1 b - I N D ∣ x ∣ 5 , formalized in the language of Buss, i.e. with ⌊ x / 2 ⌋ but without the M S P function ⌊ x / 2 y ⌋ , does not prove that every nontrivial divisor of a power of 2 is even. It follows that this theory proves neither N P = c o N P nor S 2 0 .

Probability estimates for Shor's algorithm

Let N be a (large) positive integer, let b be an integer satisfying 1&lt; b &lt; N that is relatively prime to N, and let r be the order of b modulo N. Finally, let QC be a quantum computer whose input register has the size specified in Shor's original description of his order-finding algorithm. In this paper, we analyze the probability that a single run of the quantum component of the algorithm yields useful information---a nontrivial divisor of the order sought. We prove that when Shor's algorithm is implemented on QC, then the probability P of obtaining a (nontrivial) divisor of $r$ exceeds $.7$ whenever N \ge 2^{11} and r\ge 40, and we establish that $.7736$ is an asymptotic lower bound for $P$. When $N$ is not a power of an odd prime, Gerjuoy has shown that $P$ exceeds 90 percent for $N$ and $r$ sufficiently large. We give easily checked conditions on N and r for this 90 percent threshold to hold, and we establish an asymptotic lower bound for $P$ of 2\Si(4\pi)/\pi ~0.9499 in this situation. More generally, for any nonnegative integer $q$, we show that when QC(q) is a quantum computer whose input register has $q$ more qubits than does QC, and Shor's algorithm is run on QC(q), then an asymptotic lower bound on P is 2\Si(2^{q+2}\pi)/\pi (if N is not a power of an odd prime). Our arguments are elementary and our lower bounds on P are carefully justified.

An inequality concerning the elasticity of Krull monoids with divisor class group $\mathbb{Z}_p$

Let p be a prime integer and M a Krull monoid with divisor class group $\mathbb{Z}_p$ . We represent by S the set of nontrivial divisor classes of $\mathbb{Z}_p$ which contain prime divisors. We present a new inequality for the elasticity of M (denoted ρ (M)) which is dependent on the cardinality of S and argue that this inequality is the best possible. If M as above has | S| = 3, then it is known that $\rho(M)\in \{p/2, p/3,\ldots ,p/(\lfloor p/3\rfloor + 1)\}$ , but for large p, not all the values in this containment set can be realized. For each | S| = 3, we produce a submonoid $\widetilde{\mathcal{B}}(\mathbb{Z}_p,S)$ of $\mathcal{B}(\mathbb{Z}_p,S)$ such that $$ \{\,\rho(\widetilde{\mathcal{B}}(\mathbb{Z}_p,S))\,\mid \, S\subseteq \mathbb{Z}_p-\{0\}\mbox{ and }\mid S\mid =3\}=\{p/2, p/3,\ldots ,p/(\lfloor p/3\rfloor + 1)\}. $$ Keywords: Krull monoid, Block monoid, Elasticity of factorization Mathematics Subject Classification (2000): 20M14, 20D60, 13F05