scholarly journals On the order of a modulo n, on average

2016 ◽  
Vol 12 (08) ◽  
pp. 2073-2080 ◽  
Author(s):  
Sungjin Kim
Keyword(s):  
Number Theory ◽  
Positive Constant ◽  
Acta Arith ◽  
Multiplicative Order ◽  
Orders Of Elements

Let [Formula: see text] be an integer. Denote by [Formula: see text] the multiplicative order of [Formula: see text] modulo integer [Formula: see text]. We prove that there is a positive constant [Formula: see text] such that if [Formula: see text], then [Formula: see text] where [Formula: see text] It was known for [Formula: see text] in [P. Kurlberg and C. Pomerance, On a problem of Arnold: The average multiplicative order of a given integer, Algebra Number Theory 7 (2013) 981–999] in which they refer to [F. Luca and I. E. Shparlinski, Average multiplicative orders of elements modulo [Formula: see text], Acta Arith. 109(4) (2003) 387–411.

On a problem of Erdös related to common factor differences

2019 ◽  
Vol 15 (05) ◽  
pp. 1059-1068
Author(s):  
Andrew Bremner
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Common Factor ◽  
Difference Set ◽  
Acta Arith ◽  
Factor Difference

Let [Formula: see text] be a positive integer. The factor-difference set [Formula: see text] of [Formula: see text] is the set of absolute values [Formula: see text] of the differences between the factors of any factorization of [Formula: see text] as a product of two integers. Erdős and Rosenfeld [The factor–difference set of integers, Acta Arith. 79(4) (1997) 353–359] ask whether for every positive integer [Formula: see text] there exist integers [Formula: see text] such that [Formula: see text], and prove this is true when [Formula: see text]. Urroz [A note on a conjecture of Erdős and Rosenfeld, J. Number Theory 78(1) (1999) 140–143] shows the result true for [Formula: see text]. The ideas of this paper can be extended, and here, we show the result true for [Formula: see text] by proving there are infinitely many sets of four integers with four common factor differences.

scholarly journals Artin's Primitive Root Conjecture – A Survey

Integers ◽  
2012 ◽  
Vol 12 (6) ◽  
Author(s):  
Pieter Moree
Keyword(s):  
Number Theory ◽  
Primitive Root ◽  
Number Fields ◽  
Elementary Number Theory ◽  
Open Problems ◽  
General Audience ◽  
Elementary Number ◽  
Artin's Conjecture ◽  
Multiplicative Order ◽  
Artin Primitive Root Conjecture

Abstract.One of the first concepts one meets in elementary number theory is that of the multiplicative order. We give a survey of the literature on this topic emphasizing the Artin primitive root conjecture (1927). The first part of the survey is intended for a rather general audience and rather colloquial, whereas the second part is intended for number theorists and ends with several open problems. The contributions in the survey on `elliptic Artin' are due to Alina Cojocaru. Wojciec Gajda wrote a section on `Artin for K-theory of number fields,' and Hester Graves (together with me) on `Artin's conjecture and Euclidean domains.'

scholarly journals On Zudilin's $q$-Question about Schmidt's Problem

10.37236/2231 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Victor J. W. Guo ◽  
Jiang Zeng
Keyword(s):  
Number Theory ◽  
Recent Result ◽  
Acta Arith ◽  
New Approach

We propose an elemantary approach to Zudilin's $q$-question about Schmidt's problem [Electron. J. Combin. 11 (2004), #R22], which has been solved in a previous paper [Acta Arith. 127 (2007), 17--31]. The new approach is based on a  $q$-analogue of our recent result in  [J. Number Theory 132 (2012), 1731--1740] derived from $q$-Pfaff-Saalschütz identity.

scholarly journals Analytic Erdös-Turán conjectures and Erdös-Fuchs theorem

2005 ◽  
Vol 2005 (23) ◽  
pp. 3767-3780
Author(s):  
L. Haddad ◽  
C. Helou ◽  
J. Pihko
Keyword(s):  
Number Theory ◽  
Power Series ◽  
Formal Power Series ◽  
Positive Constant ◽  
Additive Number Theory ◽  
Natural Numbers ◽  
Additive Number ◽  
Bounded Below ◽  
Formal Power

We consider and study formal power series, that we call supported series, with real coefficients which are either zero or bounded below by some positive constant. The sequences of such coefficients have a lot of similarity with sequences of natural numbers considered in additive number theory. It is this analogy that we pursue, thus establishing many properties and giving equivalent statements to the well-known Erdös-Turán conjectures in terms of supported series and extending to them a version of Erdös-Fuchs theorem.

scholarly journals On the approximation of algebraic numbers by algebraic integers

1963 ◽  
Vol 3 (4) ◽  
pp. 408-434 ◽  
Author(s):  
K. Mahler
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Number Field ◽  
Algebraic Number ◽  
Positive Constant ◽  
Algebraic Number Field ◽  
Algebraic Numbers ◽  
Algebraic Integers ◽  
Image Position ◽  
Rational Integral

In his Topics in Number Theory, vol. 2, chapter 2 (Reading, Mass., 1956) W. J. LeVeque proved an important generalisation of Roth's theorem (K. F. Roth, Mathematika 2,1955, 1—20).Let ξ be a fixed algebraic number, σ a positive constant, and K an algebraic number field of degree n. For κ∈K denote by κ(1), …, κ(n) the conjugates of κ relative to K, by h(κ) the smallest positive integer such that the polynomial has rational integral coefficients, and by q(κ) the quantity

Multiplicative Number Theory I

2006 ◽  
Author(s):  
Hugh L. Montgomery ◽  
Robert C. Vaughan
Keyword(s):  
Number Theory
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