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We correct an error in the paper of Z. Shen, S. Li, and J. Zhang published in [4]. In addition, we give an answer to a question posed by the authors. [ABSTRACT FROM AUTHOR]

The article focuses on the number theory of Gaussian integers. Within the complex numbers, the analogues of the integers are the Gaussian integers, those complex numbers whose real and imaginary parts are both integers. There is a theory of divisibility, including greatest common divisors, and the purpose of this article is to present a new gcd algorithm for Gaussian integers. The standard algorithm is a straightforward extension of the Euclidean algorithm for ordinary integers. The gcd algorithm is better suited to paper-and-pencil computation, and it is less error-susceptible than the standard one. Another attractive feature is that it is based on a simple parity argument. The basic divisibility definitions for Gaussian integers are simply restatements of those for ordinary integers. There are many interesting results which can be proved using parity arguments. The gcd algorithm establishes that any two Gaussian integers have a greatest common divisor, and it is interesting to see that this result has an odd-even proof.

It is proved that the set of those natural numbers which cannot be written as n-Ω( n) is of positive lower density. Here Ω( n) is the number of the prime power divisors of n. This is a refinement of a theorem of F. Luca. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce and study the concept of a hyperideal-based zero-divisor graph associated with a general hyperring. This is a generalized version of the zero-divisor graph associated with a commutative ring. For any general hyperring R having a hyperideal I, the I-based zero-divisor graph Γ(I)(R) associated with R is the simple graph whose vertices are the elements of R∖I having their hyperproduct in I, and two distinct vertices are joined by an edge when their hyperproduct has a non-empty intersection with I. In the first part of the paper, we concentrate on some general properties of this graph related to absorbing elements, while the second part is dedicated to the study of the I-based zero-divisor graph associated to the general hyperring Zn of the integers modulo n, when n=2pmq, with p and q two different odd primes, and fixing the hyperideal I. [ABSTRACT FROM AUTHOR]

The idempotent divisor graph of a commutative ring R is a graph with a vertex set in R* = R-{0}, where any distinct vertices x and y are adjacent if and only if x.y = e, for some non-unit idempotent element e^2 = e ∈ R, and is denoted by Л(R). Our goal in this work is to transform the planar idempotent divisor graph after coloring its regions into optical art by depending on the reflection of vertices, edges, and planes on the x or y-axes. That is, we achieve Op art solely through pure mathematics in this paper. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study the Laplacian eigenvalues of the extension of the zero divisor graph, Γ e (Z n) , for some particular values of n. We characterize the values of n that give the equality of the spectral radius and the second-smallest eigenvalue of Γ e (Z n). Finding Wiener index of Γ e (Z n) in terms of its Laplacian eigenvalues is another objective of this paper. [ABSTRACT FROM AUTHOR]

The prime graph Γ ⁢ (G) of a finite group 퐺 (also known as the Gruenberg–Kegel graph) has as its vertices the prime divisors of | G | , and p ⁢ - ⁢ q is an edge in Γ ⁢ (G) if and only if 퐺 has an element of order p ⁢ q . Since their inception in the 1970s, these graphs have been studied extensively; however, completely classifying the possible prime graphs for larger families of groups remains a difficult problem. For solvable groups, such a classification was found in 2015. In this paper, we go beyond solvable groups for the first time and characterize the prime graphs of a more general class of groups we call pseudo-solvable. These are groups whose composition factors are either cyclic or isomorphic to A 5 . The classification is based on two conditions: the vertices { 2 , 3 , 5 } form a triangle in Γ ̄ ⁢ (G) or { p , 3 , 5 } form a triangle for some prime p ≠ 2 . The ideas developed in this paper also lay the groundwork for future work on classifying and analyzing prime graphs of more general classes of finite groups. [ABSTRACT FROM AUTHOR]

For a connected graph G (V , E) a clique S ⊆ V (G) is twin-free if every pair of elements of S have distinct closed neighborhoods and the number of elements in a twin-free clique of maximum cardinality is called twin-free clique number of G. The annihilator graph A G (R) of a commutative and unital ring R is a graph whose vertices are all non-zero zero-divisors of R and there is an edge between two distinct vertices a , b if and only if ann (a) ∪ ann (b) is properly contained in ann (a b). In this paper, twin-free clique number of A G (R) is computed and as an application the strong metric dimension of A G (R) is characterized. Among other things, for a reduced ring R , the forcing strong metric dimension of A G (R) is given. [ABSTRACT FROM AUTHOR]

Consider G to be a finite group and p to be a prime divisor of the order | G | in the group G. The main aim of this paper is to prove that the outcome in a recent paper of A. Laradji is true in the case of a p-constrained group. We observe that the generalization of the concept of Navarro's vertex for an irreducible character in a p-constrained group G is generally undefined. We illustrate this with a suitable example. Let ϕ ∈ I r r (G) have a positive height, and let there be an anchor group A ϕ . We prove that if the normalizer N G (A ϕ) is p-constrained, then O p ´ (N G (A ϕ)) ≠ { 1 G } , where O p ´ (N G (A ϕ)) is the maximal normal p ´ subgroup of N G (A ϕ) . We use character theoretic methods. In particular, Clifford theory is the main tool used to accomplish the results. [ABSTRACT FROM AUTHOR]

For a commutative ring R with unity, the cozero-divisor graph denoted by Γ'(R), is an undirected simple graph whose vertex set is the set of all non-zero and non-unit elements of R. Two distinct vertices x and y are adjacent if and only if x does not belong to the ideal Ry and y does not belong to Rx. The cozero-divisor graph on the ring of integers modulo n is a generalized join of its induced sub graphs all of which are null graphs. This property of the cozero-divisor graph on Zn is used in finding its distance spectrum. In this paper, the distance matrix of the cozero-divisor graph on the ring of integers modulo n is discovered and the general method is discussed to find its distance spectrum, for any value of n. Also, the distance spectrum of this graph is explored for some values of n, by means of the vertex weighted distance matrix of the co-proper divisor graph of n. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the vertex Padmakar–Ivan index of H -generalized join of graphs. As a consequence, we find the vertex Padmakar–Ivan index of the lexicographic product of graphs, the ideal-based zero-divisor graph and the zero-divisor graph of rings. In addition, we obtain the vertex Padmakar–Ivan index of the ideal-based zero-divisor graph and the zero-divisor graph of the ring of integers modulo n. [ABSTRACT FROM AUTHOR]

This paper explores the concept of multiset dimensions (Mdim) of compressed zero-divisor graphs (CZDGs) associated with rings. The authors investigate the interplay between the ring-theoretic properties of a ring R and the associated compressed zero-divisor graph. An undirected graph consisting of a vertex set Z (R E) \ { [ 0 ] } = R E \ { [ 0 ] , [ 1 ] } , where R E = { [ x ]   : x ∈ R } and [ x ] = { y ∈ R   :   a n n (x) = a n n (y) } is called a compressed zero-divisor graph, denoted by Γ E R . An edge is formed between two vertices [ x ] and [ y ] of Z (R E) if and only if [ x ] [ y ] = [ x y ] = [ 0 ] , that is, iff x y = 0 . For a ring R , graph G is said to be realizable as Γ E R if G is isomorphic to Γ E R . We classify the rings based on Mdim of their associated CZDGs and obtain the bounds for the Mdim of the compressed zero-divisor graphs. We also study the Mdim of realizable graphs of rings. Moreover, some examples are provided to support our results. Notably, we discuss the interconnection between Mdim, girth, and diameter of CZDGs, elucidating their symmetrical significance. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and Z (R) be its zero-divisors set. The zero-divisor graph of R , denoted by Γ (R) , is an undirected graph with vertex set Z (R) ∗ = Z (R) ∖ { 0 } and two distinct vertices a and b are adjacent if and only if a b = 0. In this paper, for n = p R q S where p and q are primes (p < q) and R and S are positive integers, we calculate the A α spectrum of the graphs Γ (ℤ n). [ABSTRACT FROM AUTHOR]

In this paper, we consider the simultaneous Pell equations and where is a positive integer and is squarefree and has at most three prime divisors. We obtain the necessary and sufficient conditions that the above simultaneous Pell equations have positive integer solutions by using only the elementary methods of factorization, congruence, the quadratic residue and fundamental properties of Lucas sequence and the associated Lucas sequence. Moreover, we prove that these simultaneous Pell equations have at most one solution in positive integers. When a solution exists, assuming the positive solutions of the Pell equation are and with odd, then the only solution of the system is given by or or or . In this paper, we consider the simultaneous Pell equations and where is a positive integer and is squarefree and has at most three prime divisors. We obtain the necessary and sufficient conditions that the above simultaneous Pell equations have positive integer solutions by using only the elementary methods of factorization, congruence, the quadratic residue and fundamental properties of Lucas sequence and the associated Lucas sequence. Moreover, we prove that these simultaneous Pell equations have at most one solution in positive integers. When a solution exists, assuming the positive solutions of the Pell equation are and with odd, then the only solution of the system is given by or or or. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the interplay between the algebraic properties of a skew Poincaré–Birkhoff–Witt extesion ring A = σ (R) 〈 x 1 , ... , x n 〉 and the graph-theoretic properties of its zero-divisor graph. We are interested in studying the diameter of the zero-divisor graph of skew PBW extension rings. Among other results, we give a complete characterization of the possible diameters of Γ (A) in terms of the diameter of  Γ (R). [ABSTRACT FROM AUTHOR]

In this paper, we obtain the Harary index and the hyper-Wiener index of the H-generalized join of graphs and the generalized corona product of graphs. As a consequence, we deduce some of the results in (Das et al. in J. Inequal. Appl. 2013:339, 2013) and (Khalifeh et al. in Comput. Math. Appl. 56:1402–1407, 2008). Moreover, we calculate the Harary index and the hyper-Wiener index of the ideal-based zero-divisor graph of a ring. [ABSTRACT FROM AUTHOR]

For a finite undirected looped graph G ˚ , the universal adjacency matrix U (G ˚) is a linear combination of the adjacency matrix A (G ˚) , the degree matrix D (G ˚) , the identity matrix I and the all-ones matrix J , that is U (G ˚) = α A (G ˚) + β D (G ˚) + γ I + η J , where α , β , γ , η ∈ ℝ and α ≠ 0. For a finite commutative ring R with unity, the looped zero divisor graph Γ ˚ (R) is an undirected graph with the set of all nonzero zero divisors of R as vertices and two vertices (not necessarily distinct) x and y are adjacent if and only if x y = 0. In this paper, we study some structural properties of Γ ˚ (R) by defining an equivalence relation on its vertex set. Then we obtain the universal adjacency eigenpairs of Γ ˚ (R) , and as a consequence several spectra like the adjacency, Seidel, Laplacian, signless Laplacian, normalized Laplacian, generalized adjacency and convex linear combination of the adjacency and degree matrix of Γ ˚ (R) can be obtained in a unified way. Moreover, we get the structural properties and the universal adjacency eigenpairs of the looped zero divisor graph of a reduced ring in a simpler form. [ABSTRACT FROM AUTHOR]

For a given finite commutative ring R with 1 ≠ 0 , one may associate a graph which is called the total graph of R. This graph has R as the vertex set and its two distinct vertices x and y are adjacent exactly whenever x + y is a zero-divisor of R. In this paper, we give necessary and sufficient conditions for two classes of total graphs to be Cohen–Macaulay. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with unity, and let Γ (R) denote the comaximal graph of R. The comaximal graph Γ (R) has vertex set as R, and any two distinct vertices x, y of Γ (R) are adjacent if R x + R y = R. Let Γ 2 (R) denote the induced subgraph of Γ (R) on the set of all nonzero non-unit elements of R, and any two distinct vertices x, y of Γ 2 (R) are adjacent if R x + R y = R. In this paper, we study the graphical structure as well the adjacency spectrum of Γ 2 (ℤ n) , where n ≥ 4 is a non-prime positive integer, and ℤ n is the ring of integers modulo n. We show that for a given non-prime positive integer n with D number of positive proper divisors, the eigenvalues of Γ 2 (ℤ n) are 0 with multiplicity n − φ (n) − D − 1 , and remaining eigenvalues are contained in the spectrum of a symmetric D × D matrix. We further calculate the rank and nullity of Γ 2 (ℤ n). We also determine all the eigenvalues of Γ 2 (ℤ n) whenever Γ 2 (ℤ n) is a bipartite graph. Finally, apart from determining certain structural properties of Γ 2 (ℤ n) , we conclude the paper by determining the metric dimension of Γ 2 (ℤ n). [ABSTRACT FROM AUTHOR]

Topological complexity naturally appears in the motion planning in robotics. In this paper we consider the problem of finding topological complexity of real Grassmann manifolds $G_k(\mathbb {R}^{n})$. We use cohomology methods to give estimates on the zero-divisor cup-length of $G_k(\mathbb {R}^{n})$ for various $2\leqslant k , which in turn give us lower bounds on topological complexity. Our results correct and improve several results from Pavešić (Proc. Roy. Soc. Edinb. A 151 (2021), 2013–2029). [ABSTRACT FROM AUTHOR]

Our previous papers introduce topological notions of normal crossings symplectic divisor and variety, show that they are equivalent, in a suitable sense, to the corresponding geometric notions, and establish a topological smoothability criterion for normal crossings symplectic varieties. The present paper constructs a blowup, a complex line bundle, and a logarithmic tangent bundle naturally associated with a normal crossings symplectic divisor and determines the Chern class of the last bundle. These structures have applications in constructions and analysis of various moduli spaces. As a corollary of the Chern class formula for the logarithmic tangent bundle, we refine Aluffi's formula for the Chern class of the tangent bundle of the blowup at a complete intersection to account for the torsion and extend it to the blowup at the deepest stratum of an arbitrary normal crossings divisor. [ABSTRACT FROM AUTHOR]

Let R be a finite commutative ring with identity, I be an ideal of R and J (R) denotes the Jacobson radical of R. The ideal-based zero-divisor graph Γ I (R) of R is a graph with vertex set V (Γ I (R)) = { x ∈ R − I : x y ∈ I , for some y ∈ R − I } in which distinct vertices x and y are adjacent if and only if x y ∈ I. In this paper, we determine the diameter, girth of Γ J (R) (R). Specifically, we classify all finite commutative nonlocal rings for which Γ J (R) (R) is perfect. Furthermore, we discuss about the planarity, outerplanarity, genus and crosscap of Γ J (R) (R) and characterize all of them. [ABSTRACT FROM AUTHOR]

In this paper, we consider the sum of divisors d of n such that n / d is a power of 2 and derive a new decomposition for the number of plane partitions of n in terms of binomial coefficients as a sum over partitions of n. In this context, we introduce a new combinatorial interpretation of the number of plane partitions of n. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of classifying commutative rings according to the genus number of its associating zero-divisor graphs. The zero-divisor graph of R, where R is a commutative ring with nonzero identity, denoted by Γ (R) , is the undirected graph whose vertices are the nonzero zero-divisors of R, and the distinct vertices x and y are adjacent if and only if x y = 0 . Here, we classify the local rings with genus three zero-divisor graphs. [ABSTRACT FROM AUTHOR]

A divisor B of a nonzero polynomial A, defined over the prime field of two elements, is unitary (resp. bi-unitary) if g c d (B , A / B) = 1 (resp. g c d u (B , A / B) = 1) , where g c d u (B , A / B) denotes the greatest common unitary divisor of B and A / B . We denote by σ * * (A) the sum of all bi-unitary monic divisors of A. A polynomial A is called a bi-unitary superperfect polynomial over F 2 if the sum of all bi-unitary monic divisors of σ * * (A) equals A. In this paper, we give all bi-unitary superperfect polynomials divisible by one or two irreducible polynomials over F 2 . We prove the nonexistence of odd bi-unitary superperfect polynomials over F 2 . [ABSTRACT FROM AUTHOR]

Let ℛ be commutative ring and Z ∗ (ℛ) be the set of all non-zero zero divisors of ℛ. Then Γ (ℛ) is said to be the zero divisor graph if and only if a ⋅ b = 0 where a , b ∈ V (Γ (ℛ)) and (a , b) ∈ E (Γ (ℛ)). Graph energy ℰ (Γ (ℛ)) is defined as the sum of the absolute eigenvalues of the adjacency matrix ℳ (Γ (ℛ)) , then ℰ (Γ (ℛ)) = ∑ i = 1 n | λ i |. Wiener index W (Γ (ℛ)) is defined as the sum of all distance between pairs of vertices a and b , then W (Γ (ℛ)) = ∑ a , b ∈ V (Γ (ℛ)) d (a , b). In this paper, we compute the graph energy from the adjacency matrix of the zero divisor graph and the Wiener index from the zero divisor graph associated with commutative rings. [ABSTRACT FROM AUTHOR]

For a connected graph G(V, E) a set of vertices S ⊆ V (G) resolves the graph G, and S is a resolving set ofG, if every vertex is uniquely determined by its vector of distances to the vertices of S. A resolving set S of minimum cardinality is a metric basis forG, and the number of elements in the resolving set of minimum cardinality is the metric dimension ofG. Let R be a commutative ring with non-zero identity. The annihilator graph of R, denoted by AG(R), is the (undirected) graph whose vertex set is the set of all non-zero zero-divisors of R and two distinct vertices x and y are adjacent if and only if a n n R (x y) ≠ a n n R (x) ∪ a n n R (y) . In this paper, the metric dimension of the complement of AG(R) is studied and some metric dimension formulae for this graph are given. [ABSTRACT FROM AUTHOR]

The power graph is a special type undirected simple graph of a finite group G which has the group elements as its vertex set and two distinct vertices x , y ∈ G are adjacent if one is non-negative integral power of other. Vertex labeling of graph Γ is a process of assigning integers to all its vertices subject to certain conditions. In simple words, vertex (edge) labeling is a function of all vertices (edges) to set of labels. Frequently, integers are used in labeling of vertices and edges. A graph G is said to be divisor graph if all vertices of graph can be labeled with positive integers such that any two distinct vertices x , y ∈ G are adjacent if and only if either x | y or y | x. In this research paper, we prove that every power graph of finite group is always a divisor graph, but converse is not true. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and I be an ideal of R. The zero-divisor graph of R with respect to I, denoted by ΓI(R), is the graph whose vertices are the set {x ∈ R∖I|xy ∈ I for some y ∈ R∖I} with distinct vertices x and y are adjacent if and only if xy ∈ I. The cozero-divisor graph Γ′(R) of R is the graph whose vertices are precisely the non-zero, non-unit elements of R and two distinct vertices x and y are adjacent if and only if x∉yR and y∉xR. In this paper, we introduced and investigated a new generalization of the cozero-divisor graph Γ′(R) of R denoted by ΓI″(R). In fact, ΓI″(R) is a dual notion of ΓI(R). [ABSTRACT FROM AUTHOR]

Let Q (x) be a positive definite integral quadratic form with the determinant D being squarefree, and r (n , Q) denote the number of representations of n by the quadratic form Q. In this paper, we apply the Hardy-Littlewood-Kloosterman circle method to derive the asymptotic formula for the shifted convolution sum of the divisor function d (n) and Fourier coefficients r (n , Q). With more efforts, our method should have a number of applications for other multiplicative functions. [ABSTRACT FROM AUTHOR]

Let R be an associative unital ring, and let f ∈ R [ x ] . We say that f is a left annihilating content (AC) polynomial if f = af1 for some a ∈ R and f 1 ∈ R [ x ] with l R [ x ] (f 1) = 0 . The ring R is called a left EM-ring if each f ∈ R [ x ] is a left AC polynomial. In this paper, it is shown that R is a left EM-ring if and only if R is a left McCoy ring, and for each finitely generated right ideal I of R, there is an element a ∈ R and a finitely generated right ideal J of R with l R (J) = 0 and I = aJ. If R is a left duo right Bezout ring, then R is a left EM-ring and has property (A). For a unique product monoid G, we show that if R is a reversible left EM-ring, then the monoid ring R [ G ] is also a left EM-ring. Additionally, for a reversible right Noetherian ring R, we prove that R, R [ x ] , R [ x , x − 1 ] , and R [ [ x ] ] are all simultaneously left EM-rings. Finally, we give an application of left EM-rings (resp. strongly left EM-rings) in studying the graph of zero-divisors of polynomial rings (resp. power series rings). [ABSTRACT FROM AUTHOR]

In this paper, we study mean values for exponential sums of divisor functions. We improve previous results of [M. Pandey, Moment estimates for the exponential sum with higher divisor functions, C. R. Math. Acad. Sci. Paris  360 (2022) 419–424]. [ABSTRACT FROM AUTHOR]

Let f (z) be a holomorphic cusp form of weight κ for the full modular group S L 2 (Z). Denote its n -th normalized Fourier coefficient by λ f (n). Let τ k (n) denote that k -th divisor function with k ≥ 4. In this paper, we consider the shifted convolution sum ∑ n ≤ X τ k (n) λ f (n + h). We succeed in obtaining a non-trivial upper bound, which is uniform in the shift parameter h. [ABSTRACT FROM AUTHOR]

In this paper, it is proved that the group generated by Bass units contains a subgroup of finite index in the group of central units \mathcal {Z}(\mathcal {U}(\mathbb {Z}G)) of the integral group ring \mathbb {Z}G for a subgroup closed monomial group G with the property that every cyclic subgroup of order not a divisor of 4 or 6 is subnormal in G. If G is a generalized strongly monomial group, then it is also shown that the group generated by generalized Bass units contains a subgroup of finite index in \mathcal {Z}(\mathcal {U}(\mathbb {Z}G)). Furthermore, for a generalized strongly monomial group G, the rank of \mathcal {Z}(\mathcal {U}(\mathbb {Z}G)) is determined. The formula so obtained is in terms of generalized strong Shoda pairs of G. [ABSTRACT FROM AUTHOR]

Let RG be the group ring of the group G over a ring R and let Z*(RG) be the collection of all non-zero zero divisors in a finite group ring RG. And, for x ∈ Z*(RG), ann(x) = {r ∈ RG/rx = 0}. The annihilator graph of RG denoted as AG(RG), and is defined as the graph whose vertex set is the elements of non-zero zero divisors in RG and two distinct vertices vx and vy are adjacent if and only if ann(x) ∩ ann(y) ̸= {0}. In this paper we try to characterize the properties of annihilator graphs in group rings. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the perfect zero-divisor graphs and chordal zero-divisor graphs (its complement) of ordered sets. These results are applied to the zero-divisor graphs of finite reduced rings, the comaximal ideal graphs of rings, the annihilating ideal graphs of rings, the intersection graphs of ideals of rings, and the intersection graphs of subgroups of groups. In fact, it is shown that these graphs associated with a commutative ring R with identity can be effectively studied via the zero-divisor graph of a specially constructed poset from R. [ABSTRACT FROM AUTHOR]

Let ℒ be a bounded lattice and α : ℒ → ℒ be its automorphism. In this paper, we study zero‐divisor graph of ℒ with respect to an automorphism α. It is a simple undirected graph and denoted by Γα(ℒ). Some combinatorial structures such as coloring, diameter and girth were given for Γα(ℒ). [ABSTRACT FROM AUTHOR]

The concept of zero-divisor graphs of rings is widely used for establishing relationships between the properties of graphs and the properties of the underlying ring. The zero-divisor graph of a ring is generalized to the k-zero-divisor hypergraph of a ring R for k ∈ N , which is denoted by H k (R) . This paper is an endeavor to discuss some properties of zero-divisor hypergraphs. We determine the diameter and girth of H k (R) whenever R is reduced. Also, we characterize all commutative rings R for which H k (R) is in some known class of graphs. Further, we obtain certain necessary conditions for H k (R) to be a Hamilton Berge cycle and a flag-traversing tour. Moreover, we answer a case of the question raised by Eslahchi et al. [15]. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring. The cozero-divisor graph of R denoted by Γ'(R) is a graph with the vertex set W*(R), where W*(R) is the set of all non-zero and non-unit elements of R, and two distinct vertices x and y are adjacent if and only if x/∈ Ry and y/∈ Rx. In this paper, we investigate when the cozero-divisor graph is a line graph. We completely present all commutative rings which their cozero-divisor graphs are line graphs. Also, we study when the cozero-divisor graph is the complement of a line graph. [ABSTRACT FROM AUTHOR]

In this paper, using a deep result on the existence of primitive divisors of Lehmer numbers due to Y. Bilu, G. Hanrot and P. M. Voutier, we first give an explicit formula for all positive integer solutions of the Diophantine equation (x-d)²+x²+(x+d)² = yn (*), when n is an odd prime and d = pr, p > 3, a prime. So this improves the results of the papers of A. Koutsianas and V. Patel and A. Koutsianas. Secondly, under the assumption of our first result, we prove that (*) has at most one solution (x, y). Next, for a general d, we prove the following two results: (i) if every odd prime divisor q of d satisfies q ≢ ±1 (mod 2n), then (*) has only the solution (x, y, d, n) = (21, 11, 2, 3), and (ii) if n > 228000 and d > 8√2, then all solutions (x, y) of (*) satisfy yn < 23/2d³. [ABSTRACT FROM AUTHOR]

Let R be a non-commutative ring, and I be an ideal of R. In this paper, we generalize the definition of the zero-divisor graph of R with respect to I, and define several generalized zero-divisor graphs of R with respect to I. In this paper, we investigate the ring-theoretic properties of R and the graph-theoretic properties of all the generalized zero-divisor graphs. We study some basic properties of these generalized zero-divisor graphs related to the connectedness, the diameter and the girth. We also investigate some properties of these generalized zero-divisor graphs with respect to primal ideals. [ABSTRACT FROM AUTHOR]

In this paper, we prove the asymptotic growth rate of the summatory function of the pairwise maxima of the generalized divisor function d k (n), for a fixed positive integer k ≥ 2. This result generalizes previous results of Kátai, Erdős and Hall on the local behaviour of divisor function on short intervals. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and I be a proper ideal of a commutative ring R. The ideal-based triple zero-divisor graph of a commutative ring is a graph, denoted by TZΓI(R), with the vertex set TZI(R) = {a ∈ R : there existsb,c ∈ Rsuch thatabc ∈ I,ab∉I,bc∉I,ac∉I} and two vertices a,b are adjacent if and only if there is a c such that abc ∈ I,ab∉I,bc∉I,ac∉I. In this paper, we discuss the connectedness, diameter, girth of TZΓI(R). We classify all finite commutative rings for which TZ(ΓI(R)) is either complete, unicyclic or split graph. Also, we characterize all finite commutative rings for which TZΓI(R) is perfect. Finally, we classify all finite commutative rings for which TZΓI(R) has genus at most one. [ABSTRACT FROM AUTHOR]

AbstractFor a simple graph G with edge ideal I(G), we study the N -graded Betti numbers in the linear strand of the minimal free resolution of I(Γ(Zn)), where Γ(Zn) is the zero divisor graph of the ring Zn . We present sharp bounds for the Betti numbers of Γ(Zn) and characterize the graphs attaining these bounds, thereby establishing the correct equality case for one of the results of the earlier published paper (Theorem 4.5, S. Pirzada and S. Ahmad, On the linear strand of edge ideals of some zero divisor graphs, Commun. Algebra 51(2) (2023) 620–632). Also, we present homological invariants of the edge rings of Γ(Zn) for n=p2q and pqr, with primes p

In this paper, we investigate the square of the normalized Fourier coefficients of the primitive cusp forms f and its symmetric-lift at integers with a fixed number of distinct prime divisors, and present asymptotic formulas for them in short intervals. [ABSTRACT FROM AUTHOR]