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In 1848 Ch. Hermite formulated a question on periodic representation of cubic numbers. In this paper we introduce a new Jacobi–Perron type algorithm that provides periodicity for all cubic irrationalities in the input and prove the periodicity. This is the first subtractive algorithm for which the algebraic periodicity is proven. The algorithm is designed for the totally real case, providing the answer to Hermite's original question in this case. [ABSTRACT FROM AUTHOR]

The graph a K m , m ∇ C n is named the generalized double-wheel graph. A graph G is said to be M-integral (resp. A-integral, D-integral, D L -integral or D Q -integral) if all eigenvalues of its matrix M (resp. adjacency matrix A(G) , distance matrix D(G) , distance Laplacian matrix D L (G) or distance signless Laplacian matrix D Q (G) ) are integers. In this paper, we completely determine all D-integral, D L -integral and D Q -integral generalized double-wheel graphs respectively. [ABSTRACT FROM AUTHOR]

In 2018, Jungin Lee [5] gave a necessary and sufficient condition for a diagonal quadratic form ∑ i = 1 m a i X i 2 where a i ∈ Z for all i, 1 ≤ i ≤ m for representing all 2 × 2 matrices over Z . In this paper, we will consider the imaginary quadratic field Q (- 7) . Its ring of integers O is a principal ideal domain. Q (- 7) is the only imaginary quadratic field such that O is a principal ideal domain and 2 is a product of two distinct primes in O (upto units). With O as above, in this paper we give a necessary and sufficient condition for a diagonal quadratic form a 1 X 1 2 + a 2 X 2 2 + a 3 X 3 2 where a 1 , a 2 , a 3 ∈ O to represent all 2 × 2 matrices over O . [ABSTRACT FROM AUTHOR]

Let γ 0 = 5 - 1 2 = 0.618 ... . We prove that, for any ε > 0 and any trigonometric polynomial f with frequencies in the set { n 2 : N ⩽ n ⩽ N + N γ 0 - ε } , the inequality ‖ f ‖ 4 ≪ ε - 1 / 4 ‖ f ‖ 2 holds, which makes a progress on a conjecture of Cilleruelo and Córdoba. We also present a connection between this conjecture and the conjecture of Ruzsa which asserts that, for any ε > 0 , there is C (ε) > 0 such that each positive integer N has at most C (ε) divisors in the interval [ N 1 / 2 , N 1 / 2 + N 1 / 2 - ε ] . [ABSTRACT FROM AUTHOR]

Geometric calculus is a special case of the Non-Newtonian Calculus introduced by Grossman and Katz (Non-Newtonian calculus, Lee Press, Pigeon Cove, 1972). Also, it is a more convenient calculation method for situations where the geometric increment is more meaningful than the arithmetic increment. In this study, geometric curves are defined and geometric Frenet–Serret formulas for these curves are presented. Furthermore, we give applications of these concepts to geometric magnetic curves. [ABSTRACT FROM AUTHOR]

We give necessary and sufficient conditions for the divisibility of two finite geometric series G n (x) = 1 + x + x 2 + ⋯ + x n - 1 over a field of characteristic zero. [ABSTRACT FROM AUTHOR]

Let k ≥ 2 be a fixed integer. We consider sums of type ∑ n 1 ⋯ n k ≤ x F (n 1 , ... , n k) , taken over the hyperbolic region { (n 1 , ... , n k) ∈ N k : n 1 ⋯ n k ≤ x } , where F : N k → C is a given function. In particular, we deduce asymptotic formulas with remainder terms for the hyperbolic summations ∑ n 1 ⋯ n k ≤ x f ((n 1 , ... , n k)) and ∑ n 1 ⋯ n k ≤ x f ([ n 1 , ... , n k ]) , involving the GCD and LCM of the integers n 1 , ... , n k , where f : N → C belongs to certain classes of functions. Some of our results generalize those obtained by the authors (Heyman and Tóth in Results Math 76(1): 22, 2021) for k = 2 . [ABSTRACT FROM AUTHOR]

We obtain bounds for the error term in the prime number theorem of the form π (x) - Li (x) ≤ 9.2211 x log (x) exp - 0.8476 log (x) for all x ≥ 2 , as well as other classical forms, improving upon the various constants and ranges compared to those in the literature. The strength and originality of our methods come from leveraging numerical results for small x in order to improve both the asymptotic and numerical bounds one obtains. We develop algorithms and formulas optimizing the conversion of both asymptotic and explicit numerical bounds from the prime counting function ψ (x) to both θ (x) and π (x) . [ABSTRACT FROM AUTHOR]

We use the estimation of the number of integers n such that ⌊nc⌋ belongs to an arithmetic progression to study the coprimality of integers in ℕc = {⌊nc⌋}n∈ℕ, c > 1, c ∉ ℕ. [ABSTRACT FROM AUTHOR]

A set of positive integers is primitive (or 1-primitive) if no member divides another. Erdős proved in 1935 that the weighted sum ∑1/(n log n) for n ranging over a primitive set A is universally bounded over all choices for A. In 1988 he asked if this universal bound is attained by the set of prime numbers. One source of difficulty in this conjecture is that ∑n−λ over a primitive set is maximized by the primes if and only if λ is at least the critical exponent τ1 ≈ 1.14. A set is k-primitive if no member divides any product of up to k other distinct members. One may similarly consider the critical exponent τk for which the primes are maximal among k-primitive sets. In recent work the authors showed that τ2 < 0.8, which directly implies the Erdős conjecture for 2-primitive sets. In this article we study the limiting behavior of the critical exponent, proving that τk tends to zero as k → ∞. [ABSTRACT FROM AUTHOR]

Weinberger [16] in 1972, proved that the ring of integers of a number field with unit rank at least 1 is a principal ideal domain if and only if it is a Euclidean domain, provided the generalised Riemann hypothesis holds. Lenstra [13], extended the notion of Euclidean domains in order to capture Dedekind domains with finite cyclic class group and proved an analogous theorem in this setup. More precisely, he showed that the class group of the ring of integers of a number field with unit rank at least 1 is cyclic if and only if it has a Euclidean ideal class, provided the generalised Riemann hypothesis holds. The aim of this paper is to show the following. Suppose that K 1 and K 2 are two Galois number fields of odd prime degree with cyclic class groups and Hilbert class fields that are abelian over Q . If K 1 K 2 is ramified over K i , then at least one K i ( i ∈ { 1 , 2 } ) must have a Euclidean ideal class. [ABSTRACT FROM AUTHOR]

Motivated by the design of satins with draft of period m and step a, we draw our attention to the lattices L (m , a) = ⟨ (1 , a) , (0 , m) ⟩ where 1 ≤ a < m are integers with gcd (m , a) = 1 . We show that the extended Euclid's algorithm applied to m and a produces a shortest no null vector of L(m, a) and that the algorithm can be used to find an optimal basis of L(m, a). We also analyze square and symmetric satins. For square satins, the extended Euclid's algorithm produces directly the two vectors of an optimal basis. It is known that symmetric satins have either a rectangular or a rombal basis; rectangular basis are optimal, but rombal basis are not always optimal. In both cases, we give the optimal basis directly in terms of m and a. [ABSTRACT FROM AUTHOR]

We obtain asymptotic formulas with remainder terms for the hyperbolic summations ∑ m n ≤ x f ((m , n)) and ∑ m n ≤ x f ([ m , n ]) , where f belongs to certain classes of arithmetic functions, (m, n) and [m, n] denoting the gcd and lcm of the integers m, n. In particular, we investigate the functions f (n) = τ (n) , log n , ω (n) and Ω (n) . We also define a common generalization of the latter three functions, and prove a corresponding result. [ABSTRACT FROM AUTHOR]

In this article, we show that certain abelian extensions K with unit rank greater than or equal to three have cyclic class group if and only if it has a Euclidean ideal class. This result improves an earlier result of Murty and Graves. One can improve this result up to unit rank 2 if one assumes the Elliott and Halberstam conjecture (see Conjecture 1 in preliminaries). These results are known under generalized Riemann hypothesis by the work of Lenstra (J Lond Math Soc 10:457–465) [see also Weinberger (Proc Symp Pure Math 24:321–332)]. [ABSTRACT FROM AUTHOR]

We consider a special type of sequence of arithmetic progressions, in which consecutive progressions are related by the property: ithterms ofjth, (j + 1)thprogressions of the sequence are multiplicative inverses of each other modulo(i + 1)thterm ofjthprogression. Such a sequence is uniquely defined for any pair of co-prime numbers. A computational problem, defined in the context of such a sequence and its generalization, is shown to be equivalent to the integer factoring problem. The proof is probabilistic. As an application of the equivalence result, we propose a method for how users securely agree upon secret keys, which are ensured to be random. We compare our method with factoring based public key cryptographic systems: RSA (Rivest et al., ACM 21, 120–126, 1978) and Rabin systems (Rabin 1978). We discuss the advantages of the method, and its potential use-case in the post quantum scenario. [ABSTRACT FROM AUTHOR]

Let n and k be positive integers such that n ≥ k + 1 and let { a i } i = 1 n be an arbitrary given strictly increasing sequence of positive integers. Let S n , k : = ∑ i = 1 n - k 1 1 cm (a i , a i + k) . Borwein [3] proved a conjecture of Erdős stating that if n ≥ 2 , then S n , 1 ≤ 1 - 1 2 n - 1 , with the equality holding if and only if a i = 2 i - 1 for 1 ≤ i ≤ n . In this paper, we first improve Borwein's upper bound by showing that S n , 1 ≤ 1 a 1 (1 - 1 2 n - 1 ) with the equality occurring if and only if a i = 2 i - 1 a 1 for all integers 1 ≤ i ≤ n . Then we use this improved upper bound to show that if n ≥ 3 , then S n , 2 ≤ 7 6 + 1 2 ⌊ n 2 ⌋ (2 3 δ n - 7 3) , with the equality holding if and only if a 1 = 1 , a 2 i = 2 i and a 2 i + 1 = 3 × 2 i - 1 for all integers 1 ≤ i ≤ ⌊ n 2 ⌋ , where δ n : = 0 if n is even, and 1 if n is odd. Furthermore, we show that if n ≥ 7 , then S n , 3 ≤ 17 15 - 37 15 · 1 2 ⌊ n 3 ⌋ + ϵ n 2 ⌈ n 3 ⌉ , with equality occurring if and only if a i = i for all i ∈ { 1 , 2 , 3 } and a 3 i + 1 = 2 i + 1 (1 ≤ i ≤ ⌊ n - 1 3 ⌋) , a 3 i + 2 = 5 × 2 i - 1 (1 ≤ i ≤ ⌊ n - 2 3 ⌋) and a 3 i + 3 = 3 × 2 i (1 ≤ i ≤ ⌊ n 3 ⌋ - 1) , where ϵ n = 0 if 3 ∣ n , 1 if n ≡ 1 (mod 3) and 9 5 if n ≡ 2 (mod 3) . We also present a tight upper bound for S n , 3 if n ∈ { 4 , 5 , 6 } . [ABSTRACT FROM AUTHOR]

We call a positive integer N a 4-perfect number if the sum of all divisors of N is equal to 4N. In this paper, we give some sufficient conditions such that an odd integer exactly divisible by 3 2 cannot be 4-perfect. Moreover, we refine Euler's structure theorem on odd 4-perfect numbers given by Broughan and Zhou in the case where N is exactly divisible by 3 2 . [ABSTRACT FROM AUTHOR]

In this paper, we give upper bounds for the gcd counting function(which is an analogue for the notion of gcd in the context of holomorphic maps) in various settings. As applications, we obtain analytic dependence of entire functions from the second main theorem and multiplicative dependence under the fundamental conjecture for entire curves. [ABSTRACT FROM AUTHOR]

In recent years, we have witnessed numerous new results on divisibility properties concerning binomial sums. Many conjectures and results were proposed by Z.-W. Sun. Some of these conjectures were proved by himself, and also by some scholars such as V.J.W. Guo, G.S. Mao, B. He, and Y.P. Mu. Motivated by their work, we study the WZ method and its applications to prove divisibility properties of binomial sums. In this paper, we propose a method which can be used to explore more such divisibility properties. Additionally, we give some generalized divisibility properties which can imply those divisibility properties established by Z.-W. Sun and B. He. [ABSTRACT FROM AUTHOR]

Smith showed in 1875 that if $${n \geqq 1}$$ is an integer and $${G := {({\rm gcd}(i, j))}_{1 \leqq i, j \leqq n}}$$ is the $${n \times n}$$ matrix having gcd( i, j) as its i, j-entry for all integers i and j between 1 and n, then $${{\rm det}(G) = {\prod_{k=1}^{n}}\varphi(k)}$$ , where $${\varphi}$$ is the Euler's totient function. We show that if $${n \geqq 2}$$ is an integer and $${H := {({\rm gcd}(i, j))}_{2 \leqq i, j \leqq n}}$$ is the $${(n-1) \times (n-1)}$$ matrix having gcd( i, j) as its i, j-entry for all integers i and j between 2 and n, then We also calculate the determinants of the matrices $${{(f({\rm gcd}(x_{i}, x_{j})))}_{1 \leqq i, j \leqq n}}$$ and $${{(f({\rm lcm}(x_{i},x_{j})))}_{1\leqq i, j\leqq n}}$$ having f evaluated at $${{\rm gcd}(x_{i}, x_{j})}$$ and $${{\rm lcm}(x_{i}, x_{j})}$$ as their ( i, j)-entries, respectively, where $${S = \{x_{1},\ldots, x_{n}\}}$$ is a set of distinct positive integers such that $${x_{i} > 1}$$ for all integers i with $${1 \leqq i \leqq n}$$ and $${S \cup \{1\}}$$ is factor closed (that is, $${S \cup \{1\}}$$ contains every divisor of x for any $${x \in S \cup\{1\}}$$ ). Our result answers partially an open problem raised by Ligh [18]. [ABSTRACT FROM AUTHOR]

Consider the n× n matrix with ( i, j)'th entry gcd ( i, j). Its largest eigenvalue λ and sum of entries s satisfy λ > s / n. Because s cannot be expressed algebraically as a function of n, we underestimate it in several ways. In examples, we compare the bounds so obtained with one another and with a bound from S.Hong, R.Loewy (2004). We also conjecture that λ > 6π nlog n for all n. If n is large enough, this follows from F.Balatoni (1969). [ABSTRACT FROM AUTHOR]

Let $$\alpha \in (1/2,1)$$ be fixed. We prove that for all sufficiently large T, where we can choose $$c_\alpha = 0.18 (2\alpha -1)^{1-\alpha }$$ . The same result has already been obtained by Montgomery, with a smaller value for $$c_\alpha $$ . However, our proof, which uses a modified version of Soundararajan's 'resonance method' together with ideas of Hilberdink, is completely different from Montgomery's. This new proof also allows us to obtain lower bounds for the measure of those $$t \in [0,T]$$ for which $$|\zeta (\alpha +it)|$$ is of the order mentioned above. [ABSTRACT FROM AUTHOR]

We discuss the verification for isolated singular solutions and non-isolated singular solutions with maximum rank deficiency of general nonlinear systems which have all the relevant partial derivatives. Using the border system technique, we present a deflation algorithm to compute verified error bound for such a singular solution of the given system or for a solution of a slight perturbed system, where the solution of the slight perturbed system is near the above singular solution of the given system. By adding certain components of the solution of certain linear system, our algorithm decreases the dimension of the deflation system in each step. Numerical experiments show the performance of our algorithm. [ABSTRACT FROM AUTHOR]