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Andrews introduced the singular overpartition function C ‾ k , i ( n ) which counts the number of overpartitions of n in which no part is divisible by k and only parts ≡ ± i ( mod k ) may be overlined. In this article, we study the divisibility properties of C ‾ 4 k , k ( n ) and C ‾ 6 k , k ( n ) by arbitrary powers of 2 and 3 for infinite families of k. For an infinite family of k, we prove that C ‾ 4 k , k ( n ) is almost always divisible by arbitrary powers of 2. We also prove that C ‾ 6 k , k ( n ) is almost always divisible by arbitrary powers of 3 for an infinite family of k. Using a result of Ono and Taguchi on nilpotency of Hecke operators, we find infinite families of congruences modulo arbitrary powers of 2 satisfied by C ‾ 4 ⋅ 2 α , 2 α ( n ) and C ‾ 4 ⋅ 3 ⋅ 2 α , 3 ⋅ 2 α ( n ) .

In this paper, we study properties of the coefficients appearing in the $q$-series expansion of $\prod_{n\ge 1}[(1-q^n)/(1-q^{pn})]^\delta$, the infinite Borwein product for an arbitrary prime $p$, raised to an arbitrary positive real power $\delta$. We use the Hardy--Ramanujan--Rademacher circle method to give an asymptotic formula for the coefficients. For $p=3$ we give an estimate of their growth which enables us to partially confirm an earlier conjecture of the first author concerning an observed sign pattern of the coefficients when the exponent $\delta$ is within a specified range of positive real numbers. We further establish some vanishing and divisibility properties of the coefficients of the cube of the infinite Borwein product. We conclude with an Appendix presenting several new conjectures on precise sign patterns of infinite products raised to a real power which are similar to the conjecture we made in the $p=3$ case., Comment: 24 pages; an appendix with several new conjectures added; to appear in the Ramanujan Journal; dedicated to the memory of Richard Allen Askey

We investigate the emergence of spanning structures in sparse pseudo-random $k$-uniform hypergraphs, using the following comparatively weak notion of pseudo-randomness. A $k$-uniform hypergraph $H$ on $n$ vertices is called $(p,\alpha,\epsilon)$-pseudo-random if for all (not necessarily disjoint) vertex subsets $A_1,\dots, A_k{\subseteq} V(H)$ with $|A_1|\cdots |A_k|{\geq}\alpha n^{k}$ we have $$e(A_1,\dots, A_k)=(1\pm\epsilon)p |A_1|\cdots |A_k|.$$ For any linear $k$-uniform $F$ we provide a bound on $\alpha=\alpha(n)$ in terms of $p=p(n)$ and $F$, such that (under natural divisibility assumptions on $n$) any $k$-uniform $\big(p,\alpha, o(1)\big)$-pseudo-random $n$-vertex hypergraph $H$ with a mild minimum vertex degree condition contains an $F$-factor. The approach also enables us to establish the existence of loose Hamilton cycles in sufficiently pseudo-random hypergraphs and all results imply corresponding bounds for stronger notions of hypergraph pseudo-randomness such as jumbledness or large spectral gap. As a consequence, $\big(p,\alpha, o(1)\big)$-pseudo-random $k$-graphs as above contain: $(i)$ a perfect matching if $\alpha=o(p^{k})$ and $(ii)$ a loose Hamilton cycle if $\alpha=o(p^{k-1})$. This extends the works of Lenz--Mubayi, and Lenz--Mubayi--Mycroft who studied the analogous problems in the dense setting.

In this paper, we study the Pontryagin numbers of $24$ dimensional String manifolds. In particular, we find representatives of an integral basis of the String cobrodism group at dimension $24$, based on the work of Mahowald-Hopkins \cite{MH02}, Borel-Hirzebruch \cite{BH58} and Wall \cite{Wall62}. This has immediate applications on the divisibility of various characteristic numbers of the manifolds. In particular, we establish the $2$-primary divisibilities of the signature and of the modified signature coupling with the integral Wu class of Hopkins-Singer \cite{HS05}, and also the $3$-primary divisibility of the twisted signature. Our results provide potential clues to understand a question of Teichner., Comment: final version

The bankruptcy problem is to divide a homogeneous divisible good (the “estate”) between claimants, when the sum of the claims exceeds the value of the estate. When the problem is looked at from an ex-ante point of view (i.e. before the size of the estate is revealed), it is possible to formulate a notion of Pareto efficiency that is stronger than when the more common ex-post perspective is taken. Under the assumption of common beliefs, the strong notion of efficiency leads, in combination with the requirement that all claims should be fulfilled when the value of the estate is equal to the sum of the claims, to a uniquely defined division rule when utility functions for all agents are given. The resulting rule can be represented in the form of a parametric function. For the case in which all agents are equipped with the same utility function, the class of parametric functions that can be obtained in this way is characterized. In particular, it is shown that two well-known division rules for the bankruptcy problem, namely Constrained Equal Losses and Proportional Division, can be rationalized under strong Pareto efficiency by constant absolute risk aversion and constant relative risk aversion respectively.

Embora a teoria dos traços distintivos de Roman Jakobson esteja bem representada em seus trabalhos em inglês após sua imigração para os Estados Unidos, um quadro completo do desenvolvimento dessa teoria ainda está por ser feito, ainda que todos os seus trabalhos iniciais sobre o tópico, escritos em tcheco, francês e alemão já tenham sido bem examinados. Apesar de alguns desses trabalhos já terem sido vertidos para o inglês, pode haver diferenças enganosas entre os originais (escritos em outros idiomas) e essas traduções. Baseado na comparação entre as obras fonológicas de Jakobson publicadas no início dos anos de 1930 (“Z fonologie spisovné slovenštiny” e algumas entradas da enciclopédia Ottův) e suas versões presentes em seus Selected Writings, o presente artigo procura esclarecer alguns detalhes sobre a divisibilidade do fonema, a natureza paradigmática do traço distintivo e a nomenclatura e a classificação dos traços distintivos. Este estudo também procura oferecer um exemplo específico sobre como evitar anacronismos na pesquisa sobre história da linguística.

A partition of n is $${\ell }$$ -regular if none of its parts is divisible by $${\ell }$$ . Let $$b_{{\ell }}(n)$$ denote the number of $${\ell }$$ -regular partitions of n. In this paper, using the theory of Hecke eigenforms explored by J.-P. Serre, we establish exact criteria for the 3-divisibility of $$b_9(n)$$ and $$b_{27}(n)$$ .

Lucas sequences of the first and second kinds are, respectively, the integer sequences $ (U_n)_{n\geq0} $ and $ (V_n)_{n\geq0} $ depending on parameters $ a, b\in\mathbb{Z} $ and defined by the recurrence relations $ U_0 = 0 $, $ U_1 = 1 $, and $ U_n = aU_{n-1}+bU_{n-2} $ for $ n\geq2 $, $ V_0 = 2 $, $ V_1 = a $, and $ V_n = aV_{n-1}+bV_{n-2} $ for $ n\geq2 $. In this article, we obtain exact divisibility results concerning $ U_n^k $ and $ V_n^k $ for all positive integers $ n $ and $ k $. This and our previous article extend many results in the literature and complete a long investigation on this problem from 1970 to 2021.

Linear time invariant (LTI) systems are often represented by rational forms or rational matrix functions. Such forms exhibit important properties of a system, but these properties are often thought to be due to time invariance. Are there such forms for linear time-variant (LTV) systems with comparable properties, generalizing the LTI case? One major obstacle to derive them has been the lack of a divisibility theory for general block lower-triangular matrices, in particular the lack of a Euclidean algorithm or, in its wake, Smith forms and Smith-McMillan forms. But do rational forms for matrices, generalizing the time-invariant properties, exist at all? It turns out they do. The theory presented here produces representations for block lower-triangular matrices as ratios of (block) staircase (or echelon) matrices, and shows how central properties such as co-prime factorization or Bezout identities hold. Instead of the Euclidean algorithm, the theory uses a technique derived from a paper of Paul Van Dooren, namely ‘dead-beat control’. The theory finds good applications in various domains (computational efficiency, determining pre-conditioners, control theory and model reduction via generalized forms of interpolation).

The alternating descent statistic on permutations was introduced by Chebikin as a variant of the descent statistic. We show that the alternating descent polynomials on permutations, called alternating Eulerian polynomials, are unimodal via a five-term recurrence relation. We also find a quadratic recursion for the alternating major index q-analog of the alternating Eulerian polynomials. As an interesting application of this quadratic recursion, we show that $$(1+q)^{\lfloor n/2\rfloor }$$ divides $$\sum _{\pi \in {{\mathfrak {S}}}_n}q^{\mathrm{altmaj}(\pi )}$$ , where $${{\mathfrak {S}}}_n$$ is the set of all permutations of $$\{1,2,\ldots ,n\}$$ and $$\mathrm{altmaj}(\pi )$$ is the alternating major index of $$\pi $$ . This leads us to discover a q-analog of $$n!=2^{\ell }m$$ , m odd, using the statistic of alternating major index. Moreover, we study the $$\gamma $$ -vectors of the alternating Eulerian polynomials by using these two recursions and the cd-index. Further intriguing conjectures are formulated, which indicate that the alternating descent statistic deserves more work.

Several recent papers have examined a rational polyhedron Pm whose integer points are in bijection with the numerical semigroups (cofinite, additively closed subsets of the non-negative integers) containing m. A combinatorial description of the faces of Pm was recently introduced, one that can be obtained from the divisibility posets of the numerical semigroups a given face contains. In this paper, we study the faces of Pm containing arithmetical numerical semigroups and those containing certain glued numerical semigroups, as an initial step towards better understanding the full face structure of Pm . In most cases, such faces only contain semigroups from these families, yielding a tight connection to the geometry of Pm .

A finite set of prime numbers [Formula: see text] is called unavoidable with respect to [Formula: see text] if for each [Formula: see text] the sequence of integer parts [Formula: see text], [Formula: see text] contains infinitely many elements divisible by at least one prime number [Formula: see text] from the set [Formula: see text]. It is known that an unavoidable set exists with respect to [Formula: see text] and that it does not exist if [Formula: see text] is an integer such that [Formula: see text] is not square free. In this paper, we show that no finite unavoidable sets exist with respect to [Formula: see text] if [Formula: see text] is a prime number or [Formula: see text] belongs to some explicitly given arithmetic progressions, for instance, [Formula: see text] and [Formula: see text], [Formula: see text]

This paper is the first part of a new proof of decidability of the existential theory of the structure 〈 $$\mathbb{Z}$$ ; 0, 1, +, –, ≤, |〉, where | corresponds to the binary divisibility relation. The decidability was proved independently in 1976 by A.P. Bel’tyukov and L. Lipshitz. In 1977, V.I. Mart’yanov proved an equivalent result by considering the ternary GCD predicate instead of divisibility (the predicates are interchangeable with respect to existential definability). Generalizing in some sense the notion of quantifier elimination (QE) algorithm, we construct a quasi-QE algorithm to prove decidability of the positive existential theory of the structure 〈 $${{\mathbb{Z}}_{{ > 0}}}$$ ; 1, $${{\{ a{\kern 1pt} \cdot \} }_{{a \in {{\mathbb{Z}}_{{ > 0}}}}}}$$ , GCD〉. We reduce to the decision problem for this theory the decision problem for the existential theory of the structure 〈 $$\mathbb{Z}$$ ; 0, 1, +, –, ≤, GCD〉. A quasi-QE algorithm, which performs this reduction, will be constructed in the second part of the proof. Transformations of formulas are based on a generalization of the Chinese remainder theorem to systems of the form GCD(ai, bi + x) = di for every $$i \in [1..m]$$ , where ai, bi, di are some integers such that $${{a}_{i}} \ne 0$$ , $${{d}_{i}} > 0$$ .

Let $\mathcal{KC}_g ^k$ be the moduli stack of pairs $(S,C)$ with $S$ a $K3$ surface and $C\subset S$ a genus $g$ curve with divisibility $k$ in $\mathrm{Pic}(S)$. In this article we study the forgetful map $c_g^k:(S,C) \mapsto C$ from $\mathcal{KC}_g ^k$ to $\mathcal{M}_g$ for $k>1$. First we compute by geometric means the dimension of its general fibre. This turns out to be interesting only when $S$ is a complete intersection or a section of a Mukai variety. In the former case we find the existence of interesting Fano varieties extending $C$ in its canonical embedding. In the latter case this is related to delicate modular properties of the Mukai varieties. Next we investigate whether $c_g^k$ dominates the locus in $\mathcal{M}_g$ of $k$-spin curves with the appropriate number of independent sections. We are able to do this only when $S$ is a complete intersection, and obtain in these cases some classification results for spin curves., v2: post-final version. Various enhancements in Sec.4 (including new subsection 4.4 on maximal variation) that will not appear in the published version, to appear in Boll. Unione Mat. Ital

Let f ∈ Z [ x ] and l ∈ N . Consider the set of all ( a 0 , a 1 , … , a l ) ∈ N l + 1 with a i a i + 1 and f ( a i ) | f ( a i + 1 ) for all 0 ≤ i ≤ l − 1 . We say that f satisfies the gap principle of order l if lim ⁡ a l / a 0 = ∞ as a 0 → ∞ . We also define the gap order of f ( x ) to be the smallest positive integer l such that f ( x ) satisfies the gap principle of order l. If such l does not exist, we say that f ( x ) does not satisfy the gap principle. In this article, we prove a conjecture by Chan, Choi and Lam that f ( x ) does not satisfy the gap principle if and only if f ( x ) is in the form of f ( x ) = A ( B x + C ) n for some A , B , C ∈ Z . Moreover, we completely determine the gap order of any polynomial. We will show that if f ( x ) is not in the form of A ( B x + C ) n , then f ( x ) has gap order 2 if f ( x ) is a quadratic polynomial or a power of a quadratic polynomial; and has gap order 1 otherwise.

For any odd prime p, we construct an infinite family of imaginary quadratic fields whose class numbers are divisible by p. We give a corollary that settles Iizuka’s conjecture for the case n=1 and p>2.

Objectives: Teaching mathematical theorems in Indian classroom settings usually focuses on explaining the statement of the theorem followed by methods of giving its proof. This hardly enables students to learn how to write a new theorem. Even teachers, at times, express their inability to write a new theorem. The objective of this article is to present a situational casestudy explaining the general processes involved to facilitate writing a new mathematical theorem. Methods: The methodology followed for coming up with a new mathematical theorem is based on careful observation of events around and recognizing patterns thereof. The emphasis is on summing up facts at different levels of observations, in order to evolve a concise theorem. Findings: From a small observation of the arrangements of the numbers on the keypads of a mobile phone or a calculator, a general mathematical theorem is evolved. This is an example for the pedagogy to be followed for writing a mathematical theorem, which can be useful both for the teachers and the students to self-educate themselves in new situations and casestudies. Novelty: This article formulates an innovative pedagogy using the mathematical concepts of subtraction, addition, divisibility, pattern recognition and generalization & randomization of event to evolve a mathematical theorem. The transformation in statement of the theorem with addition of more and more facts leading to its newer versions is the main feature of this article. Keywords: Mathematics Education; Mathematical Theorem; Innovative Pedagogy; Pattern recognition; Teaching Methods

In this text, the exclusive exponential Diophantine equation px + (p + 1)y= z2such that the sum of integer powers and of two consecutive prime numbers engrosses a square is examined or estimating enormous integer solutions by exploiting the fundamental notion of Mathematics and the speculation of divisibility or all possibilities of x + y = 1, 2, 3, 4..

Summary In this paper problems 14, 15, 29, 30, 34, 78, 83, 97, and 116 from [6] are formalized, using the Mizar formalism [1], [2], [3]. Some properties related to the divisibility of prime numbers were proved. It has been shown that the equation of the form p 2 + 1 = q 2 + r 2, where p, q, r are prime numbers, has at least four solutions and it has been proved that at least five primes can be represented as the sum of two fourth powers of integers. We also proved that for at least one positive integer, the sum of the fourth powers of this number and its successor is a composite number. And finally, it has been shown that there are infinitely many odd numbers k greater than zero such that all numbers of the form 22 n + k (n = 1, 2, . . . ) are composite.

The Fibonacci sequence, whose first terms are f0; 1; 1; 2; 3; 5; : : :g, is generated using the recursive formula Fn+2 = Fn+1 + Fn with F0 = 0 and F1 = 1. This sequence is one of the most famous integer sequences because of its fascinating mathematical properties and connections with other fields such as biology, art, and music. Closely related to the Fibonacci sequence is the Lucas sequence. The Lucas sequence, whose first terms are f2; 1; 3; 4; 7; 11; : : :g, is generated using the recursive formula Ln+2 = Ln+1 + Ln with L0 = 2 and L1 = 1. In this paper, patterns in the prime factors of sums of powers of Fibonacci and Lucas numbers are examined. For example, F2 3n+4 + F2 3n+2 is even for all n 2 N0. To prove these results, techniques from modular arithmetic and facts about the divisibility of Fibonacci and Lucas numbers are utilized. KEYWORDS: Fibonacci Sequence; Lucas Sequence; Modular Arithmetic; Divisibility Sequence

We study the divisibility properties of the partition function associated with the eighth-order mock theta function $$V_0(q)$$ , introduced by Gordon and McIntosh. We obtain congruences modulo powers of 2 for certain coefficients of the partition function, akin to Ramanujan’s partition congruences. Further, we also present several infinite families of congruences modulo 13, 25, and 27.

We continue the research of an extension (Formula presented.) of the divisibility relation to the Stone-Čech compactification (Formula presented.). First we prove that ultrafilters we call prime actually possess the algebraic property of primality. Several questions concerning the connection between divisibilities in (Formula presented.) and nonstandard extensions of (Formula presented.) are answered, providing a few more equivalent conditions for divisibility in (Formula presented.). Results on uncountable chains in (Formula presented.) are proved and used in a construction of a well-ordered chain of maximal cardinality. Probably the most interesting result is the existence of a chain of type (Formula presented.) in (Formula presented.). Finally, we consider ultrafilters without divisors in (Formula presented.) and among them find the greatest class.

We continue the research of the relation $\hspace{1mm}\widetilde{\mid}\hspace{1mm}$ on the set $\beta {\mathbb{N}}$ of ultrafilters on ${\mathbb{N}}$, defined as an extension of the divisibility relation. It is a quasiorder, so we see it as an order on the set of $=_\sim$-equivalence classes, where ${\cal F}=_\sim{\cal G}$ means that ${\cal F}$ and ${\cal G}$ are mutually $\hspace{1mm}\widetilde{\mid}\hspace{1mm}$-divisible. Here we introduce a new tool: a relation of congruence modulo an ultrafilter. We first recall the congruence of ultrafilters modulo an integer and show that $=_\sim$-equivalent ultrafilters do not necessarily have the same residue modulo $m\in {\mathbb{N}}$. Then we generalize this relation to congruence modulo an ultrafilter in a natural way. After that, using iterated nonstandard extensions, we introduce a stronger relation, which has nicer properties with respect to addition and multiplication of ultrafilters. Finally, we also introduce a strengthening of $\hspace{1mm}\widetilde{\mid}\hspace{1mm}$ and show that it also behaves well in relation to the congruence relation., Comment: 1 figure

Andrews introduced the partition function $\overline {C}_{k, i}(n)$ , called the singular overpartition function, which counts the number of overpartitions of n in which no part is divisible by k and only parts $\equiv \pm i\pmod {k}$ may be overlined. We prove that $\overline {C}_{6, 2}(n)$ is almost always divisible by $2^k$ for any positive integer k. We also prove that $\overline {C}_{6, 2}(n)$ and $\overline {C}_{12, 4}(n)$ are almost always divisible by $3^k$ . Using a result of Ono and Taguchi on nilpotency of Hecke operators, we find infinite families of congruences modulo arbitrary powers of $2$ satisfied by $\overline {C}_{6, 2}(n)$ .

Have you ever wondered how certain divisibility tests were even happened upon? Every student of number theory has at one time or another been exposed to certain divisibility tests for small primes ...

Let $k \geq 1$ be a cube-free integer with $k \equiv 1 \pmod {9}$ and $\gcd(k, 7\cdot 571)=1$. In this paper, we prove the existence of infinitely many triples of imaginary quadratic fields $\mathbb{Q}(\sqrt{d})$, $\mathbb{Q}(\sqrt{d+1})$ and $\mathbb{Q}(\sqrt{d+k^2})$ with $d \in \mathbb{Z}$ such that the class number of each of them is divisible by $3$. This affirmatively answers a weaker version of a conjecture of Iizuka \cite{iizuka-jnt}., To appear in Acta Arithmetica

Divisibility of finite sums of power of integers such as , , and has been developed by authors in some books and articles. In this article will be discussion residual of result division divisible by two by using divisibility concept and formulate it based on residual of result division that is listed into tables.

Let A be an infinite dimensional stably finite unital simple separable C*-algebra. Let B ⊂ A be a stably (centrally) large subalgebra in A such that B is m-almost divisible (m-almost divisible, weakly (m, n)-divisible). Then A is 2(m + 1)-almost divisible (weakly m-almost divisible, secondly weakly (m, n)-divisible).

In this paper we confirm three conjectures of Z.-W. Sun on determinants. We first show that any odd integer $n>3$ divides the determinant $$\left|(i^2+dj^2)\left(\frac{i^2+dj^2}n\right)\right|_{0\le i,j\le (n-1)/2},$$ where $d$ is any integer and $(\frac{\cdot}n)$ is the Jacobi symbol. Then we prove some divisibility results concerning $|(i+dj)^n|_{0\le i,j\le n-1}$ and $|(i^2+dj^2)^n|_{0\le i,j\le n-1}$, where $d\not=0$ and $n>2$ are integers. Finally, for any odd prime $p$ and integers $c$ and $d$ with $p\nmid cd$, we determine completely the Legendre symbol $(\frac{S_c(d,p)}p)$, where $S_c(d,p):=|(\frac{i^2+dj^2+c}p)|_{1\le i,j\le(p-1)/2}$., 14 pages, accepted by Linear and Multilinear Algebra

Given a vectorial function ${F}:\mathbb {F}_{2}^{{n}} \mapsto \mathbb {F}_{2}^{{m}}$ , the indicator $1_{{\mathcal {G}}_{{F}}}$ of its graph ${\mathcal{ G}}_{{F}}=\{({x},{F}({x})); {x}\in \mathbb {F}_{2}^{{n}}\}$ allows to express the algebraic degree of $F$ in a simple way. Exploiting the formula, obtained in a previous article, for the graph indicator of a composite function ${G}\circ {F}$ , that involves only a sum of products of $1_{{\mathcal {G}}_{{F}}}$ and $1_{{\mathcal {G}}_{{G}}}$ , we deduce the exact expression as well as bounds on the algebraic degree of ${G}\circ {F}$ , whose efficiency comes from the fact that the algebraic degree of the product of two Boolean functions is bounded above by the sum of their algebraic degrees, while for a composition, it is bounded above by their product. One of these bounds, that depends on the algebraic degrees of $G$ and $1_{{\mathcal {G}}_{{F}}}$ , is tight, general, simple, and most often efficient (for the case where it is not efficient, we give an improved bound, that is a little more complex). As far as we know, it is the first efficient upper bound ever found, that works without any condition on the vectorial functions. It provides a new criterion for the choice of S-boxes in block ciphers. It implies as a corollary a known bound assuming the divisibility of the Walsh transform values by a power of 2. It gives a better view why this latter bound works. All the bounds generalize to more than two functions and this represents also an improvement over the state of the art. When $F$ is a permutation, our expression of the algebraic degree of ${G}\circ {F}$ simplifies into a formula involving the algebraic degrees of the products of a coordinate function of $G$ and coordinate functions of ${F}^{-1}$ . This implies another known bound showing that the algebraic degree of ${F}^{-1}$ has more impact on that of ${G}\circ {F}$ than that of $F$ itself. Our approach by graph indicators gives a more complete explanation to this interesting fact. Our results include all the known efficient bounds as particular cases, and clarify the reasons why they work. We also deduce the exact expression of the algebraic degree of the composition of any number of functions, leading to a bound that is much more efficient than what we obtain by applying the known bound several times. We also obtain two bounds on the algebraic degree of ${G} \circ {F}$ , where $F$ is a permutation, given the divisibility by powers of 2 of some Walsh transform values of component functions of $F$ and their sums with a coordinate function of $G$ . We compare all the bounds of this kind obtained so far and show how they are complementary, and we study the generalizations of all (known and new) bounds of this kind to the composition of more than two functions.

In this paper, we examine properties of the tensor powers of quantum mappings Φ. In particular, we review positivity properties of unitary and nonunitary qubit mappings Φ⊗2. For arbitrary finite-dimensional systems, we present the relationship between the positive and completely positive divisibility of dynamical mappings $$ {\Phi}_t^{\otimes 2} $$ and Φt. A criterion of annihilation of entanglement by an arbitrary qubit mapping Φ⊗2 is found.

In Pacific J. Math. 292 (2018), 223-238, Shareshian and Woodroofe asked if for every positive integer $n$ there exist primes $p$ and $q$ such that, for all integers $k$ with $1 \leq k \leq n-1$, the binomial coefficient $\binom{n}{k}$ is divisible by at least one of $p$ or $q$. We give conditions under which a number $n$ has this property and discuss a variant of this problem involving more than two primes. We prove that every positive integer $n$ has infinitely many multiples with this property.

In this manuscript we discuss the notion of (statistical) independence embedded in its historical context. We focus in particular on its appearance and role in number theory, concomitantly exploring the intimate connection of independence and the famous Gaussian law of errors. As we shall see, this at times requires us to go adrift from the celebrated Kolmogorov axioms, which give the appearance of being ultimate ever since they have been introduced in the 1930s. While these insights are known to many a mathematician, we feel it is time for both a reminder and renewed awareness. Among other things, we present the independence of the coefficients in a binary expansion together with a central limit theorem for the sum-of-digits function as well as the independence of divisibility by primes and the resulting, famous central limit theorem of Paul Erdős and Mark Kac on the number of different prime factors of a number $$n\in{\mathbb{N}}$$ n ∈ N . We shall also present some of the (modern) developments in the framework of lacunary series that have its origin in a work of Raphaël Salem and Antoni Zygmund.

We introduce the concept of a quasiresiduated lattice and prove that every lattice effect algebra can be organized into a commutative quasiresiduated lattice with divisibility. Also conversely, every such a lattice can be converted into a lattice effect algebra and every lattice effct algebra can be reconstructed form its assigned quasiresiduated lattice. We apply this method also for lattice pseudoeffect algebras introduced recently by Dvurecenskij and Vetterlein. We show that every good lattice pseudoeffect algebra can be organized into a (possibly non-commutative) quasiresiduated lattice with divisibility and conversely, every such a lattice can be converted into a lattice pseudoeffect algebra. Moreover, also a good lattice pseudoeffect algebra can be reconstructed from the assigned quasiresiduated lattice.

Lucas sequence of the first kind is an integer sequence $(U_n)_{n\geq0}$ which depends on parameters $a, b\in\mathbb{Z}$ and is defined by the recurrence relation $U_0 = 0$, $U_1 = 1$, and $U_n = aU_{n-1}+bU_{n-2}$ for $n\geq2$. In this article, we obtain exact divisibility results concerning $U_n^k$ for all positive integers $n$ and $k$. This extends many results in the literature from 1970 to 2020 which dealt only with the classical Fibonacci and Lucas numbers $(a = b = 1)$ and the balancing and Lucas-balancing numbers $(a = 6, b = -1)$.

Let D be an integral domain, F+(D) (resp., f+(D)) be the set of nonzero (resp., nonzero finitely generated) ideals of D, R1 = f+(D) ∪ {(0)}, and R2 = F+(D) ∪ {(0)}. Then (Ri, ⊕, ⊗) for i = 1, 2 is a commutative semiring with identity under I ⊕ J = I + J and I ⊗ J = IJ for all I, J ∈ Ri. In this paper, among other things, we show that D is a Prüfer domain if and only if every ideal of R1 is a k-ideal if and only if R1 is Gaussian. We also show that D is a Dedekind domain if and only if R2 is a unique factorization semidomain if and only if R2 is a principal ideal semidomain. These results are proved in a more general setting of star operations on D.

A finite group $G$ is called $\psi$-divisible if $\psi(H)|\psi(G)$ for any subgroup $H$ of $G$, where $\psi(H)$ and $\psi(G)$ are the sum of element orders of $H$ and $G$, respectively. In this paper, we extend a result provided in [10], by classifying the finite groups whose all subgroups are $\psi$-divisible. Since the existence of $\psi$-divisible groups is related to the class of square-free order groups, we also study the sum of element orders and the $\psi$-divisibility property of ZM-groups. In the end, we introduce the concept of $\psi$-normal divisible group, i.e. a group for which the $\psi$-divisibility property is satisfied by all its normal subgroups. Using simple and quasisimple groups, we are able to construct infinitely many $\psi$-normal divisible groups which are neither simple nor nilpotent., Comment: 10 pages

In this article, we give explicit formulas for the $p$-adic valuations of the Fibonomial coefficients ${p^a n \choose n}_F$ for all primes $p$ and positive integers $a$ and $n$. This is a continuation from our previous article extending some results in the literature, which deal only with $p = 2,3,5,7$ and $a = 1$. Then we use these formulas to characterize the positive integers $n$ such that ${pn \choose n}_F$ is divisible by $p$, where $p$ is any prime which is congruent to $\pm 2 \pmod{5}$., updated and submitted

It is shown that the positive existential theory of the structure ℱS = (S−1F[t];=,F, 0, 1,+, |, f ↦ tf), where f ↦ tf is the multiplication by t map, S is non-empty a finite set of irreducible polynomials, and F is a finite field of odd characteristic, is undecidable.

In this paper we relate t-designs to a forbidden configuration problem in extremal set theory. Let 1_t 0_l denote a column of t 1's on top of l 0's. We assume t>l. Let q. (1_t 0_l) denote the (t+l)xq matrix consisting of t rows of q 1's and l rows of q 0's. We consider extremal problems for matrices avoiding certain submatrices. Let A be a (0,1)-matrix forbidding any (t+l)x(\lambda+2) submatrix (\lambda+2). (1_t 0_l) . Assume A is m-rowed and only columns of sum t+1,t+2,... ,m-l are allowed to be repeated. Assume that A has the maximum number of columns subject to the given restrictions. Assume m is sufficiently large. Then A has each column of sum 0,1,... ,t and m-l+1,m-l+2,..., m exactly once and, given the appropriate divisibility condition, the columns of sum t+1 correspond to a t-design with block size t+1 and parameter \lambda and there are no other columns. The proof derives a basic upper bound on the number of columns of A by a pigeonhole argument and then a careful argument, for large m, reduces the bound by a substantial amount down to the value given by design based constructions. We extend in a few directions., Comment: arXiv admin note: text overlap with arXiv:1909.07580

In this paper we introduce a class of sequences connected with the m–ary partition function and investigate their congruence properties. In particular, we get facts about the sequences of m–ary partitions ( b m ( n ) ) n ∈ N and m–ary partitions with no gaps ( c m ( n ) ) n ∈ N . We prove, for example, that for any natural number 2 h ≤ m each residue class modulo h appears infinitely many times in both sequences ( b m ( n ) ) n ∈ N and ( c m ( n ) ) n ∈ N . Moreover, we give new proofs of characterisations modulo m in terms of base–m representation of n for sequences ( b m ( n ) ) n ∈ N and ( c m ( n ) ) n ∈ N . We also present a general method of finding such characterisations modulo any power of m. Using our approach we get a description of ( b m ( n ) mod μ 2 ) n ∈ N , where μ 2 = m 2 if m is odd and μ 2 = m 2 / 2 if m is even.

In this paper we prove that the [math] -adic [math] -function that interpolates the Rankin–Selberg product of a general modular form and a CM form of higher weight divides the characteristic ideal of the corresponding Selmer group. This is one divisibility of the Iwasawa main conjecture for this [math] -adic [math] -function. We prove this conjecture using congruences between Klingen–Eisenstein series and cusp forms on the group [math] , following the strategy of recent work by C. Skinner and E. Urban. The actual argument is, however, more complicated due to the need to work with general Fourier–Jacobi expansions. This theorem is used to deduce a converse of the Gross–Zagier–Kolyvagin theorem and the [math] -adic part of the precise BSD formula in the rank one case.

This research seeks to examine, first, whether and why consumers perceive divisible versus indivisible numbers differently and, second, how such divergent perceptions influence consumer preferences for marketer-created entities associated with divisible versus indivisible numbers. Integrating insights from two different literatures—numerical cognition and loneliness—we propose and find that numbers perceived to be divisible (vs. indivisible) are viewed as having more “connections” and are therefore deemed to be less lonely. Building on these findings and the literature on compensatory consumption, we then propose and demonstrate that a temporary feeling of loneliness increases participants’ relative preference for various targets—products, attributes, and prices—associated with divisible (vs. indivisible) numbers, which are perceived to be relatively more connected and less lonely. It merits mention that our findings are triangulated across a wide variety of numbers, different product categories, and multiple operationalizations of loneliness.

We confirm three conjectures of Guo and Zeng on Franel numbers and binomial coefficients. Furthermore, we prove more refined divisibility results for a large class of polynomial sequences.

In 1974, P. Hartung proved the existence of infinitely many imaginary quadratic fields with class numbers not divisible by 3. We generalize this result to show the existence of infinitely many imag...