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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]

Two decades ago, Steven Roman, Daniel E. Loeb and Gian-Carlo Rota introduced a family of harmonic numbers in their study of harmonic logarithms. We propose to refer to those numbers as Roman harmonic numbers . With the purpose of revitalizing the study of these mathematical objects, we recall here their known properties and unveil additional ones. An integral representation, several generating relations, and a collection of sum rules involving those numbers are presented. It is also shown that higher derivatives of the Pochhammer and reciprocal Pochhammer symbols are easily expressed in terms of Roman harmonic numbers. [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]

We prove new estimates on the number of algebraic points of fixed degree and bounded height on projective spaces over a given number field. These results extend previous works of Wolfgang Schmidt [13] , Gao Xia [3] and Martin Widmer [18] . Our approach, based on zeta functions, also gives a new proof of Schanuel's theorem. [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]

Some congruences on conjectures of van Hamme are established. These results confirm some conjectures of Swisher. [ABSTRACT FROM AUTHOR]

For each 1 < β < 2 , let r n ( β ) be the maximal length of consecutive 0 digits in the first n digits of 1's β -expansions. We prove that for Lebesgue almost all 1 < β < 2 , lim n → ∞ ⁡ r n ( β ) / log β ⁡ n = 1 and r n ( β ) ≥ log β ⁡ n for infinitely many n , which answers two questions raised by Erdös, Joó and Komornik. [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]

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

Let m be a fifth power free integer greater than one. Let K be an algebraic number field generated by a fifth root of m over the rational number field. If m has a prime factor p congruent to −1 modulo five, the class number of K is a multiple of five. [ABSTRACT FROM AUTHOR]

Abstract: Text: Let K be an algebraic number field and the ring of integers in K. Let be the set of all elements which are sums of squares in and the minimal number of squares necessary to represent −1 in . Let be the smallest positive integer t such that every element in is a sum of t squares in . In this note, for , where are two distinct positive square-free integers, we show that and if , then . Video: For a video summary of this paper, please click here or visit http://youtu.be/F6UHHgzfdzA. [Copyright &y& Elsevier]

Abstract: In the paper, the author establishes several integral representations and properties of Stirling numbers of the first kind. [Copyright &y& Elsevier]

Abstract: We use Artin–Schreier base change to construct counterexamples to a Kummer–Vandiver type question for function fields. [Copyright &y& Elsevier]

Abstract: Let C be an algebraically closed field of positive characteristic p and complete with respect to a non-archimedean absolute value and a discrete -submodule. Suppose there exists an -basis of Λ such that and . For define the meromorphic function on C. We show that all the zeroes x of satisfy for some i. Furthermore, the number (counted with multiplicities) of zeroes for which (⁎) holds depends only on i and the p-adic expansion coefficients of k, but not on Λ. [Copyright &y& Elsevier]

Abstract: Following methods of Hayes, we state a conjectural product formula for ratios of Brumer–Stark units over real quadratic fields. [Copyright &y& Elsevier]

Abstract: Text: The isomorphism number of an F-crystal over an algebraically closed field of positive characteristic is the smallest non-negative integer such that the -th level truncation of determines the isomorphism class of . When is isoclinic, namely it has a unique Newton slope λ, we provide an efficiently computable upper bound for in terms of λ and the Hodge slopes of . This is achieved by providing an upper bound for the level torsion of introduced by Vasiu. We also check that this upper bound is optimal for many families of isoclinic F-crystals that are of special interest (such as isoclinic F-crystals of K3 type). Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=gVObAZZ1DKE. [Copyright &y& Elsevier]

Abstract: Let p be an odd prime number. For the cyclotomic -extension of a finite algebraic number field F, we denote by the maximal unramified pro-p-extension of . In this paper, using Iwasawa theory and the theory of central class fields, we give necessary conditions for to be abelian, and give sufficient conditions in certain special cases. [Copyright &y& Elsevier]

Abstract: The Hadamard product of two power series and is the power series . We define the (Hadamard) grade of a power series A to be the least number (finite or infinite) of algebraic power series, the Hadamard product of which equals A. We study and discuss this notion. [Copyright &y& Elsevier]

Abstract: We give an alternative computation of the Galois group of the maximal 2-ramified and complexified pro-2-extension of any 2-rational number field (Theorem 2), a particular case of results of Movahhedi–Nguyen Quang Do. This short Note is motivated by the paper [J. Jossey, Galois 2-extensions unramified outside 2, J. Number Theory 124 (2007) 42–76] and, at this occasion, we bring into focus some classical technics of abelian ℓ-ramification which, unfortunately, are often ignored, especially those developed by J.-F. Jaulent with the ℓ-adic class field theory, and by G. Gras in his book on class field theory, and which considerably simplify the study of such subjects; for instance, our proof of Theorem 2 generalizes the purpose of Jossey''s paper in such a way using a result of Herfort–Zalesskii. This Note is mainly an attempt of clarification about ℓ-rationality. [Copyright &y& Elsevier]

Abstract: In this paper we develop a method for determining the number of integers without large prime factors lying in a given set S. We will apply it to give an easy proof that certain sufficiently dense sets A and B always produce the expected number of “smooth” sums , , . The proof of this result is completely combinatorial and elementary. [Copyright &y& Elsevier]

Abstract: We modify an idea of Maire to construct biquadratic number fields with small root discriminants, class number one, and having an infinite, necessarily non-solvable, strictly unramified Galois extension. [Copyright &y& Elsevier]