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This document provides a summary of a special issue of the journal Mathematics, which features research papers in the field of algebraic, analytic, and computational number theory. The papers cover a wide range of topics, including generalized Lucas pseudoprimality, Fibonacci numbers, piecewise linear difference equations, properties of Euler's function, regularities in ordered semihypergroups, zero-density results for automorphic functions, properties of Fibonacci finite operator quaternions, and more. The authors of the special issue believe that the techniques used in these papers have the potential to yield new results in these areas of mathematics. [Extracted from the article]

The Hurwitz zeta-function ζ (s , α) , s = σ + i t , with parameter 0 < α ⩽ 1 is a generalization of the Riemann zeta-function ζ (s) ( ζ (s , 1) = ζ (s) ) and was introduced at the end of the 19th century. The function ζ (s , α) plays an important role in investigations of the distribution of prime numbers in arithmetic progression and has applications in special function theory, algebraic number theory, dynamical system theory, other fields of mathematics, and even physics. The function ζ (s , α) is the main example of zeta-functions without Euler's product (except for the cases α = 1 , α = 1 / 2 ), and its value distribution is governed by arithmetical properties of α. For the majority of zeta-functions, ζ (s , α) for some α is universal, i.e., its shifts ζ (s + i τ , α) , τ ∈ R , approximate every analytic function defined in the strip { s : 1 / 2 < σ < 1 } . For needs of effectivization of the universality property for ζ (s , α) , the interval for τ must be as short as possible, and this can be achieved by using the mean square estimate for ζ (σ + i t , α) in short intervals. In this paper, we obtain the bound O (H) for that mean square over the interval [ T − H , T + H ] , with T 27 / 82 ⩽ H ⩽ T σ and 1 / 2 < σ ⩽ 7 / 12 . This is the first result on the mean square for ζ (s , α) in short intervals. In forthcoming papers, this estimate will be applied for proof of universality for ζ (s , α) and other zeta-functions in short intervals. [ABSTRACT FROM AUTHOR]

Monogenity is a classical area of algebraic number theory that continues to be actively researched. This paper collects the results obtained over the past few years in this area. Several of the listed results were presented at a series of online conferences titled "Monogenity and Power Integral Bases". We also give a collection of the most important methods used in several of these papers. A list of open problems for further research is also given. [ABSTRACT FROM AUTHOR]

The first topic covered in [[1]] is built around the Gauss hypergeometric function HT ht , the Kummer hypergeometric function HT ht , and their generalization HT ht . Finally, paper [[11]] investigates certain identities associated with HT ht -binomial coefficients and HT ht -Stirling polynomials of the second kind. Hypergeometric Bernoulli numbers and polynomials connect to the hypergeometric function HT ht which is the subject of [[4]]. [Extracted from the article]

In this paper, we study the action of diamond operators on Hilbert modular forms with coefficients in a general commutative ring. In particular, we generalize a result of Chai on the surjectivity of the constant term map for Hilbert modular forms with nebentype to the setting of group ring valued modular forms. As an application, we construct certain Hilbert modular forms required for Dasgupta–Kakde's proof of the Brumer–Stark conjecture at odd primes. Since the forms required for the Brumer–Stark conjecture live on the non-PEL Shimura variety associated to the reductive group G = Res F / Q (GL 2) , as opposed to the PEL Shimura variety associated to the subgroup G ∗ ⊂ G studied by Chai, we give a detailed explanation of theory of algebraic diamond operators for G, as well as how the theory of toroidal and minimal compactifications for G may be deduced from the analogous theory for G ∗ . [ABSTRACT FROM AUTHOR]

In this article, we find some properties of certain types of entropies of a natural number. We are studying a way of measuring the "disorder" of the divisors of a natural number. We compare two of the entropies H and H ¯ defined for a natural number. An useful property of the Shannon entropy is the additivity, H S (pq) = H S (p) + H S (q) , where pq denotes tensor product, so we focus on its study in the case of numbers and ideals. We mention that only one of the two entropy functions discussed in this paper satisfies additivity, whereas the other does not. In addition, regarding the entropy H of a natural number, we generalize this notion for ideals, and we find some of its properties. [ABSTRACT FROM AUTHOR]

This article is devoted to the even entire functions, which are the exact solutions for the Laplace and diffusion equations. These functions are considered in the algebraic number field. We guess that the functions have purely real zeros in the entire complex plane. These are proposed as new connections with algebraic number theory and mathematical physics. [ABSTRACT FROM AUTHOR]

Index modulation (IM) is an attractive concept for next generation communication systems. However, as an emerging technique, there are still some challenges need to be tackled in this frontier. In this paper, we consider two aspects of IM, which are diversity and detection complexity. Specifically, we propose a linear precoding assisted index modulation (LPIM) scheme for orthogonal frequency division multiplexing (OFDM) systems. We commence by analyzing the diversity and coding gains of the proposed scheme. Moreover, a detailed codebook design criterion is proposed. Then, our LPIM codebook is designed based on the maximum diversity and coding gain criteria. In contrast to the signaling model of the existing full diversity precoder designed for OFDM, we introduce a modeling method to link the zero-valued IM symbols to the origin of a Lattice for implementing our full diversity precoder designed for OFDM-IM. Both analytical and computer simulation results are provided for characterizing the attainable performance of OFDM-LPIM, demonstrating that it is capable of achieving full diversity as well as an attractive coding gain. However, the maximum-likelihood (ML) detection complexity of OFDM-LPIM is excessive, hence a low-complexity generalized iterative residual check detector (GIRCD) is proposed, which is inspired by the existing sparse recovery algorithms. Finally, computer simulation results are provided for demonstrating that GIRCD can provide a beneficial trade-off between bit error ratio performance and complexity. [ABSTRACT FROM AUTHOR]

Despite several works on secrecy coding for fading and MIMO wiretap channels from an error probability perspective, the construction of information-theoretically secure codes over such channels remains an open problem. In this paper, we consider a fading wiretap channel model where the transmitter has only partial statistical channel state information. Our channel model includes static channels, i.i.d. block fading channels, and ergodic stationary fading with fast decay of large deviations for the eavesdropper’s channel. We extend the flatness factor criterion from the Gaussian wiretap channel to fading and MIMO wiretap channels, and establish a simple design criterion where the normalized product distance/minimum determinant of the lattice and its dual should be maximized simultaneously. Moreover, we propose concrete lattice codes satisfying this design criterion, which are built from algebraic number fields with constant root discriminant in the single-antenna case, and from division algebras centered at such number fields in the multiple-antenna case. The proposed lattice codes achieve strong secrecy and semantic security for all rates $R< C_{b}-C_{e}-\kappa $ , where $C_{b}$ and $C_{e}$ are Bob and Eve’s channel capacities, respectively, and $\kappa $ is an explicit constant gap. Furthermore, these codes are almost universal in the sense that a fixed code is good for secrecy for a wide range of fading models. Finally, we consider a compound wiretap model with a more restricted uncertainty set, and show that rates $R< \bar {C}_{b}-\bar {C}_{e}-\kappa $ are achievable, where $\bar {C}_{b}$ is a lower bound for Bob’s capacity and $\bar {C}_{e}$ is an upper bound for Eve’s capacity for all the channels in the set. [ABSTRACT FROM AUTHOR]

This paper considers space-time coding over several independently Rayleigh faded blocks. In particular, we will concentrate on giving upper bounds for the coding gain of lattice space-time codes as the number of blocks grow. This problem was previously considered in the single antenna case by Bayer-Fluckiger et al. in 2006. Crucial to their work was Odlyzko’s bound on the discriminant of an algebraic number field, as this provides an upper bound for the normalized coding gain of number field codes. In the MIMO context natural codes are constructed from division algebras defined over number fields and the coding gain is measured by the discriminant of the corresponding (noncommutative) algebra. In this paper, we will develop analogues of the Odlyzko bounds in this context and show how these bounds limit the normalized coding gain of a very general family of division algebra based space-time codes. These bounds can also be used as benchmarks in practical code design and as tools to analyze asymptotic bounds of performance as the number of independently faded blocks increases. [ABSTRACT FROM AUTHOR]

Traditional public key exchange protocols are based on algebraic number theory. In another perspective, neural cryptography, which is based on neural networks, has been emerging. It has been reported that two parties can exchange secret key pairs with the synchronization phenomenon in neural networks. Although there are various models of neural cryptography, called Tree Parity Machine (TPM), many of them are not suitable for practical use, considering efficiency and security. In this paper, we propose a Vector-Valued Tree Parity Machine (VVTPM), which is a generalized architecture of TPM models and can be more efficient and secure for real-life systems. In terms of efficiency and security, we show that the synchronization time of the VVTPM has the same order as the basic TPM model, and it can be more secure than previous results with the same synaptic depth. [ABSTRACT FROM AUTHOR]

This paper is about a generalization of celebrated Inoue’s surfaces. To each matrix M in SL(2n+1, Z) we associate a complex non-Kähler manifold TM of complex dimension n + 1. This manifold fibers over S¹ with the fiber T2n+1 and monodromy MT. Our construction is elementary and does not use algebraic number theory. We show that some of the OeljeklausToma manifolds are biholomorphic to the manifolds of type TM. We prove that if M is not diagonalizable, then TM does not admit a Kähler structure and is not homeomorphic to any of Oeljeklaus-Toma manifolds. [ABSTRACT FROM AUTHOR]

The paper deals with the class of entire functions that increase not faster than exp{γ∣z∣6/5(ln∣z∣)−1} and that, together with their first derivatives, take values from a fixed field of algebraic numbers at the points of a two-dimensional lattice of general form (in this case, the values increase not too fast). It is shown that any such functions is either a polynomial or can be represented in the form e−mαzP(eαz), where m is a nonnegative integer, P is a polynomial, and α is an algebraic number. [ABSTRACT FROM AUTHOR]

In this paper, the one-dimensional α-fractional diffusion equation is revisited. This equation is a particular case of the time- and space-fractional diffusion equation with the quotient of the orders of the time- and space-fractional derivatives equal to one-half. First, some integral representations of its fundamental solution including the Mellin-Barnes integral representation are derived. Then a series representation and asymptotics of the fundamental solution are discussed. The fundamental solution is interpreted as a probability density function and its entropy in the Shannon sense is calculated. The entropy production rate of the stochastic process governed by the α-fractional diffusion equation is shown to be equal to one of the conventional diffusion equation. [ABSTRACT FROM AUTHOR]

A Gaussian integer is a complex number whose real and imaginary parts are both integers. Meanwhile, a sequence is defined as perfect if and only if it has an ideal periodic auto-correlation function. This paper proposes a method for constructing sparse perfect Gaussian integer sequences (SPGISs) in which most of the sequence elements are zero. The proposed SPGISs are obtained by linearly combining four base sequences or their cyclic-shift equivalents using nonzero Gaussian integer coefficients of equal magnitudes. Each base sequence contains four nonzero elements belonging to the set \\pm 1, \pm j\. The number of nonzero elements of the constructed SPGISs depends on the choice of complex coefficients and cyclic shifts. However, each SPGIS has at most 16 nonzero elements, irrespective of the sequence length. A systematic investigation is performed into the properties of the SPGISs and their Fourier dual equivalents. Finally, a general expression is derived for a perfect Gaussian integer sequence (PGIS) of length $4n$, where $n$ is any positive integer and most of the sequence elements are nonzero. [ABSTRACT FROM AUTHOR]

This paper investigates the design of codes for multiple-input multiple-output (MIMO) multiple access channel (MAC). If a joint maximum-likelihood decoding is to be performed at the receiver, then every MIMO-MAC code can be regarded as a single-user code, where the minimum determinant criterion proposed by Tarokh et al. is useful for designing such codes and for upper bounding the maximum pairwise error probability (PEP), whenever the codes are of finite rate and operate in finite signal-to-noise ratio range. Unlike the case of single-user codes where the minimum determinant can be lower bounded by a fixed constant as code-rate grows, it was proved by Lahtonen et al. that the minimum determinant of MIMO-MAC codes decays as a function of the rates. This decay phenomenon is further investigated in this paper, and upper bounds for the decays of minimum determinant corresponding to each error event are provided. Lower bounds for the optimal decay are established and are based on an explicit construction of codes using algebraic number theory and Diophantine approximation. For some error profiles, the constructed codes are shown to meet the aforementioned upper bounds, hence they are optimal finite-rate codes in terms of PEPs associated with such error events. An asymptotic diversity-multiplexing gain tradeoff (DMT) analysis of the proposed codes is also given. It is shown that these codes are DMT optimal when the values of multiplexing gains are small. [ABSTRACT FROM PUBLISHER]

In this paper, we present a number field version of the celebrated result of Lenstra ( Math. Comp. 42(165) () 331-340) in 1984. Also, this result allows us to improve a result of Wikstrőm (On the l-ary GCD-algorithm in rings of integers () pp. 1189-1201). [ABSTRACT FROM AUTHOR]

Let G be a group and ω(G) = {o(g)|g ∈ G} be the set of element orders of G. Let k ∈n ω(G) and sk = |{g ∈ G |o(g) = k}|. Let nse(G) = {sk|k ∈ ω(G)}. In this paper, we prove that if G is a group and G2(4) is the Chevalley group such that nse(G) = nse(G2(4)), then G ≅ G2(4). [ABSTRACT FROM AUTHOR]

In this paper we study the groups of circular numbers and circular units in Sinnott's sense in real abelian fields with exactly four ramified primes under certain conditions. More specifically, we construct Z-bases for them in five special infinite families of cases. We also derive some results about the corresponding module of relations (in one family of cases, we show that the module of Ennola relations is cyclic). The paper is based upon the thesis [6], which builds upon the results of the paper [2]. [ABSTRACT FROM AUTHOR]

Since 1970, reversible finite automata have generated interest among researcher; but till now, we have not come across a model of reversible read only one-way finite automata which accept all regular languages, In this paper, we introduce a new model of one-way reversible finite automata inspired by the Watson-Crick complementarity relation and show that our model can accept all regular languages. We further show that our model accepts a language which is not accepted by any multi-head deterministic finite automaton. [ABSTRACT FROM AUTHOR]

In 1975, Don Zagier obtained a new version of the Kronecker limit formula for a real quadratic field which involved an interesting function F (x) which is now known as the Herglotz function. As demonstrated by Zagier, and very recently by Radchenko and Zagier, F (x) satisfies beautiful properties which are of interest in both algebraic number theory as well as in analytic number theory. In this paper, we study F k , N (x) , an extension of the Herglotz function which also subsumes higher Herglotz function of Vlasenko and Zagier. We call it the extended higher Herglotz function. It is intimately connected with a certain generalized Lambert series as well as with a generalized cotangent Dirichlet series inspired from Krätzel's work. We derive two different kinds of functional equations satisfied by F k , N (x). Radchenko and Zagier gave a beautiful relation between the integral ∫ 0 1 log ⁡ (1 + t x) 1 + t d t and F (x) and used it to evaluate this integral at various rational as well as irrational arguments. We obtain a relation between F k , N (x) and a generalization of the above integral involving polylogarithm. The asymptotic expansions of F k , N (x) and some generalized Lambert series are also obtained along with other supplementary results. [ABSTRACT FROM AUTHOR]

The article offers information on preparation of manuscripts for the periodical and copying along with the table of contents.

A primitive completely normal element for the extension E / F of Galois fields is a generator of the multiplicative group of E, which simultaneously is normal in E / K for every intermediate field K of E / F. Letting $$F=\mathbb {F}_q$$ und $$E=\mathbb {F}_{q^n}$$ , we denote by $$P_n(q)$$ the number of primitive elements of E, and by $$PCN_n(q)$$ the number of primitive completely normal elements for E / F. We prove that $$PCN_n(q) >0$$ provided that $$\ln (2) q \ge (t(n)-1)\cdot (\ln (2)+n\ln (q))$$ , where $$t(n):=\sum _{d|n}d$$ denotes the sum of all positive divisors of n and $$\ln $$ is the natural logarithm. Based on this estimate, we are able to prove that, for every fixed n, the quotient $$PCN_n(q)/P_n(q)$$ converges to 1 as q goes to infinity. We also achieve effective versions of the first result: $$PCN_n(q)>0$$ whenever $$q\ge n^{7/2}$$ and $$n\ge 7$$ , or when $$q\ge n^3$$ and $$n\ge 37$$ . The paper is self-contained and the proofs rely only on elementary combinatorics and number theory, as well as some basic calculus. [ABSTRACT FROM AUTHOR]

We call an abelian variety over a finite field F q super-isolated if its (F q -rational) isogeny class contains a single isomorphism class. In this paper, we use the Honda-Tate theorem to characterize super-isolated ordinary simple abelian varieties by certain algebraic integers. Our main result is that for a fixed dimension g ≥ 3 , there are finitely many such varieties over all finite fields. [ABSTRACT FROM AUTHOR]

Dedekind sum first occurred naturally in Dedekind's transformation law of his eta-function. In analogy, Hardy sums are encountered in the transformation formula of the theta function. Up to now, they have many of remarkable applications in analytic number theory (Dedekind's η-function), algebraic number theory (class numbers), lattice point problems, topology and algebraic geometry. Miscellaneous arithmetical properties of these sums have been analyzed by many scholars. Recently, considering the characteristics of Hardy sums and other celebrated sums such as Ramanujan sum and Kloosterman sum, interesting and meaningful identities have been obtained. In this paper, we continue to focus on arithmetical aspects of Hardy sums and Ramanujan sum. More precisely, we consider a mean value problem of these sums and Ramanujan sum with the help of the features of Dirichlet L-functions and present some computational formulas for them. [ABSTRACT FROM AUTHOR]

Abstract In this paper we describe how to use the algorithmic methods provided by Hunter and Pohst in order to give a complete classification of number fields of degree 8 and signature (2 , 3) with absolute discriminant less than a certain bound. The choice of this bound comes from the local corrections given by prime ideals to the lower estimates for discriminants obtained with the Odlyzko–Poitou–Serre method. [ABSTRACT FROM AUTHOR]

Matrix inversion is one of the most significant operations on a matrix. For any non-singular matrix A ∈ Z n × n , the inverse of this matrix may contain countless numbers of non-integer entries. These entries could be endless floating-point numbers. Storing, transmitting, or operating such an inverse could be cumbersome, especially when the size n is large. The only square integer matrix that is guaranteed to have an integer matrix as its inverse is a unimodular matrix U ∈ Z n × n . With the property that det (U) = ± 1 , then U − 1 ∈ Z n × n is guaranteed such that U U − 1 = I , where I ∈ Z n × n is an identity matrix. In this paper, we propose a new integer matrix G ˜ ∈ Z n × n , which is referred to as an almost-unimodular matrix. With det (G ˜) ≠ ± 1 , the inverse of this matrix, G ˜ − 1 ∈ R n × n , is proven to consist of only a single non-integer entry. The almost-unimodular matrix could be useful in various areas, such as lattice-based cryptography, computer graphics, lattice-based computational problems, or any area where the inversion of a large integer matrix is necessary, especially when the determinant of the matrix is required not to equal ± 1 . Therefore, the almost-unimodular matrix could be an alternative to the unimodular matrix. [ABSTRACT FROM AUTHOR]

In this paper, we continue the work of Jang et al. and classify the ( − 3 ) -saturated indecomposable fat Hoffman graphs with an integral representation of norm 3. This gives a contribution to Problem 4 of Woo and Neumaier. [ABSTRACT FROM AUTHOR]

Define D n ( x ) = ∑ k = 0 n ( n k ) 2 x k ( x + 1 ) n − k for n = 0 , 1 , 2 , … and s n ( x ) = ∑ k = 1 n 1 n ( n k ) ( n k − 1 ) x k − 1 ( x + 1 ) n − k for n = 1 , 2 , 3 , … . Then D n ( 1 ) is the n -th central Delannoy number D n , and s n ( 1 ) is the n -th little Schröder number s n . In this paper we obtain some surprising arithmetic properties of D n ( x ) and s n ( x ) . We show that 1 n ∑ k = 0 n − 1 D k ( x ) s k + 1 ( x ) ∈ Z [ x ( x + 1 ) ] for all n = 1 , 2 , 3 , … . Moreover, for any odd prime p and p -adic integer x ≢ 0 , − 1 ( mod p ) , we establish the supercongruence ∑ k = 0 p − 1 D k ( x ) s k + 1 ( x ) ≡ 0 ( mod p 2 ) . As an application we confirm Conjecture 5.5 in [S14a] , in particular we prove that 1 n ∑ k = 0 n − 1 T k M k ( − 3 ) n − 1 − k ∈ Z for all n = 1 , 2 , 3 , … , where T k is the k -th central trinomial coefficient and M k is the k -th Motzkin number. [ABSTRACT FROM AUTHOR]

There is a large literature on the asymptotic distribution of numbers free of large prime factors, so-called smooth or friable numbers. But there is very little known about this distribution that is numerically explicit. In this paper we follow the general plan for the saddle point argument of Hildebrand and Tenenbaum, giving explicit and fairly tight intervals in which the true count lies. We give two numerical examples of our method, and with the larger one, our interval is so tight we can exclude the famous Dickman–de Bruijn asymptotic estimate as too small and the Hildebrand–Tenenbaum main term as too large. [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve defined over a number field K . Then its Mordell–Weil group E ( K ) is finitely generated: E ( K ) ≅ E ( K ) t o r × Z r . In this paper, we discuss the cyclic torsion subgroup of elliptic curves over cubic number fields. For N = 169 , 143 , 91 , 65 , 77 or 55, we show that Z / N Z is not a subgroup of E ( K ) t o r for any elliptic curve E over a cubic number field K . [ABSTRACT FROM AUTHOR]

Let F be a totally real algebraic number field, with O F the ring of algebraic integers in F , let L be an O F -lattice on a d -dimensional ( d ≥ 2 ) positive definite quadratic space V over F and let u 0 be a primitive vector in L . The main objective of this paper is to study when a is a fixed (non-unit) algebraic integer in O F and n is a positive rational integer, how the class numbers of lattices translation L + u 0 a n grow as n tends to infinity. [ABSTRACT FROM AUTHOR]

Let a , b , c be fixed coprime positive integers with min ⁡ { a , b , c } > 1 . In this paper, combining the Gel'fond–Baker method with an elementary approach, we prove that if max ⁡ { a , b , c } > 5 × 10 27 , then the equation a x + b y = c z has at most three positive integer solutions ( x , y , z ) . [ABSTRACT FROM AUTHOR]

Let χ range over the non-trivial primitive characters associated with the abelian extensions L / K of a given number field K , i.e. over the non-trivial primitive characters on ray class groups of K . Let f χ be the norm of the finite part of the conductor of such a character. It is known that | L ( 1 , χ ) | ≤ 1 2 Res s = 1 ( ζ K ( s ) ) log ⁡ f χ + O ( 1 ) , where the implied constants in this O ( 1 ) are effective and depend on K only. The proof of this result suggests that one can expect better upper bounds by taking into account prime ideals of K dividing the conductor of χ , i.e. ramified prime ideals. This has already been done only in the case that K = Q . This paper is devoted to giving for the first time such improvements for any K . As a non-trivial example, we give fully explicit bounds when K is an imaginary quadratic number field. [ABSTRACT FROM AUTHOR]

In this paper we first of all determine all possible genera of (odd and even) definite unimodular lattices over an imaginary-quadratic field. The main questions are whether the partial class numbers of lattices with given Steinitz class within one genus are equal for all occurring Steinitz classes and whether the partial masses of those partial genera are equal. We show that the answer to the first question in general is “no” by giving a counter example, while the answer to the second question is “yes” by proving a mass formula for partial masses. Finally, we determine a list of all single-class partial genera and show that a partial genus consists of only one class if and only if the whole genus consists of only one class. [ABSTRACT FROM AUTHOR]

In this paper, we study multizeta values over function fields in characteristic p . For each d ≥ 2 , we show that when the constant field has cardinality >2, the field generated by all multizeta values of depth d is of infinite transcendence degree over the field generated by all single zeta values. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the 3-rank of the ideal class group of the imaginary quadratic field Q ( 4 − 3 18 n + 3 ) is at least 3 for every positive integer n . [ABSTRACT FROM AUTHOR]

Continuing the work of Johnson (2018), we classify the pure fields and pure valued fields of dp-rank 1, up to elementary equivalence. In the process, we give a few new examples of dp-minimal fields. Specializing to the case of dp-small fields, we prove that dp-small fields and VC-minimal fields are all algebraically closed or real closed, as had been conjectured by Guingona (2014). This generalizes earlier work of Haskell and Macpherson (1994) and Macpherson et al. (2000), showing that C -minimal fields are algebraically closed, and weakly o-minimal fields are real closed. In fact, we obtain slight strengthenings of these earlier results, since we assume no compatibility between the field structure on the one hand and the C -relation or order on the other hand. We also give a new proof of Jahnke, Simon, and Walsberg's (2017) theorem that dp-minimal valued fields are henselian. Lastly, we apply the Kaplan-Scanlon-Wagner (2011) theorem (NIP fields are Artin-Schreier closed) to prove some general properties of strongly dependent valued fields. For example, strongly dependent henselian valued fields are defectless. [ABSTRACT FROM AUTHOR]

Possible transcendental nature of Euler's constant γ has been the focus of study for sometime now. One possible approach is to consider γ not in isolation, but as an element of the infinite family of generalized Euler–Briggs constants. In a recent work [6] , it is shown that the infinite list of generalized Euler–Briggs constants can have at most one algebraic number. In this paper, we study the dimension of spaces generated by these generalized Euler–Briggs constants over number fields. More precisely, we obtain non-trivial lower bounds (see Theorem 5 and Theorem 6 ) on the dimension of these spaces and consequently establish the infinite dimensionality of the space spanned. Further, we study linear and algebraic independence of these constants over the field of all algebraic numbers. [ABSTRACT FROM AUTHOR]

Text In 2014, Herzog, Kaplan and Lev introduced the critical numbers cr ( r , n ) of natural intervals [ r , n ] and determined the values of cr ( 1 , n ) and cr ( 2 , n ) for all n , and for r ≥ 3 , they determined the values of cr ( r , n ) for n ≤ ( 3 r 2 + 3 r ) / 2 and n ≥ ( 5 r 2 + r ) / 2 . In this note, we determine the remaining values of cr ( r , n ) for r ≥ 3 . Video For a video summary of this paper, please visit https://youtu.be/hZSI-Ul3K94 . [ABSTRACT FROM AUTHOR]

Let A K denote the ring of algebraic integers of an algebraic number field K = Q ( θ ) where the algebraic integer θ has minimal polynomial F ( x ) = x n + a x m + b over the field Q of rational numbers with n = m t + u , t ∈ N , 0 ≤ u ≤ m − 1 . In this paper, we characterize those primes which divide the discriminant of F ( x ) but do not divide [ A K : Z [ θ ] ] when u = 0 or u divides m ; such primes p are important for explicitly determining the decomposition of p A K into a product of prime ideals of A K in view of the well known Dedekind theorem. As a consequence, we obtain some necessary and sufficient conditions involving only a , b , m , n for A K to be equal to Z [ θ ] . [ABSTRACT FROM AUTHOR]

In this paper we deal with Diophantine equations involving products of consecutive integers, inspired by a question of Erdős and Graham. [ABSTRACT FROM AUTHOR]

In this paper we work on indivisibility of the class numbers of real quadratic function fields. We find an explicit expression for a lower bound of the density of real quadratic function fields (with constant field F ) whose class numbers are not divisible by a given prime ℓ . We point out that the explicit lower bound of such a density we found only depends on the prime ℓ , the degrees of the discriminants of real quadratic function fields, and the condition: either | F | ≡ 1 ( mod ℓ ) or not. [ABSTRACT FROM AUTHOR]

The goal of this paper is to generalize and refine the classical ramification theory of complete discrete valuation rings to more general valuation rings, in the case of Artin–Schreier extensions. We define refined versions of invariants of ramification in the classical ramification theory and compare them. Furthermore, we can treat the defect case. [ABSTRACT FROM AUTHOR]

In this paper, we sharpen and simplify our earlier results based on Thue's Fundamentaltheorem and use it to obtain effective irrationality measures for certain roots of particular polynomials of the form ( x − t ) n + ( x + t ) n , where n ≥ 4 is a positive integer and t is a negative integer. For n = 4 and n = 5 , we find infinitely many such algebraic numbers. [ABSTRACT FROM AUTHOR]