scholarly journals A remark on a formula for a sum of positive integer power of natural numbers

1899 ◽  
Vol 028 (2) ◽  
pp. 124-125
Author(s):  
Vilém Jung
Keyword(s):  
Positive Integer ◽  
Natural Numbers ◽  
Integer Power

scholarly journals ON THE DENSITY OF INTEGERS OF THE FORM (p−1)2−n IN ARITHMETIC PROGRESSIONS

2008 ◽  
Vol 78 (3) ◽  
pp. 431-436 ◽  
Author(s):  
XUE-GONG SUN ◽  
JIN-HUI FANG
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Asymptotic Density ◽  
Natural Numbers ◽  
Positive Proportion ◽  
Covering System

AbstractErdős and Odlyzko proved that odd integers k such that k2n+1 is prime for some positive integer n have a positive lower density. In this paper, we characterize all arithmetic progressions in which natural numbers that can be expressed in the form (p−1)2−n (where p is a prime number) have a positive proportion. We also prove that an arithmetic progression consisting of odd numbers can be obtained from a covering system if and only if those integers in such a progression which can be expressed in the form (p−1)2−n have an asymptotic density of zero.

scholarly journals On normal numbers

1982 ◽  
Vol 32 (1) ◽  
pp. 79-87 ◽  
Author(s):  
C. E. M. Pearce ◽  
M. S. Keane
Keyword(s):  
Positive Integer ◽  
Simple Proof ◽  
Normal Numbers ◽  
Integer Power ◽  
Positive Integers

AbstractSchmidt has shown that if r and s are positive integers and there is no positive integer power of r which is also a positive integer power of s, then there exists an uncountable set of reals which are normal to base r but not even simply normal to base s. We give a structurally simple proof of this result

scholarly journals Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52

2017 ◽  
Vol 15 (1) ◽  
pp. 446-458 ◽  
Author(s):  
Ebénézer Ntienjem
Keyword(s):  
Positive Integer ◽  
Modular Forms ◽  
Quadratic Forms ◽  
Modular Space ◽  
Natural Numbers ◽  
Convolution Sums ◽  
Sum Of Divisors

Abstract The convolution sum, $ \begin{array}{} \sum\limits_{{(l\, ,m)\in \mathbb{N}_{0}^{2}}\atop{\alpha \,l+\beta\, m=n}} \sigma(l)\sigma(m), \end{array} $ where αβ = 22, 44, 52, is evaluated for all natural numbers n. Modular forms are used to achieve these evaluations. Since the modular space of level 22 is contained in that of level 44, we almost completely use the basis elements of the modular space of level 44 to carry out the evaluation of the convolution sums for αβ = 22. We then use these convolution sums to determine formulae for the number of representations of a positive integer by the octonary quadratic forms $a\,(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2})+b\,(x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}),$ where (a, b) = (1, 11), (1, 13).

Independent Gödel sentences and independent sets

1975 ◽  
Vol 40 (2) ◽  
pp. 159-166
Author(s):  
A. M. Dawes ◽  
J. B. Florence
Keyword(s):  
Positive Integer ◽  
Independent Sets ◽  
Peano Arithmetic ◽  
Natural Numbers ◽  
Simple Set ◽  
Logical Notion ◽  
Infinite Set ◽  
Element Set

In this paper we investigate some of the recursion-theoretic problems which are suggested by the logical notion of independence.A set S of natural numbers will be said to be k-independent (respectively, ∞-independent) if, roughly speaking, in every correct system there is a k-element set (respectively, an infinite set) of independent true sentences of the form x ∈ S. S will be said to be effectively independent (respectively, absolutely independent) if given any correct system we can generate an infinite set of independent (respectively, absolutely independent) true sentences of the form x ∈ S.We prove that(a) S is absolutely independent ⇔S is effectively independent ⇔S is productive;(b) for every positive integer k there is a Π1 set which is k-independent but not (k + 1)-independent;(c) there is a Π1 set which is k-independent for all k but not ∞-independent;(d) there is a co-simple set which is ∞-independent.We also give two new proofs of the theorem of Myhill [1] on the existence of an infinite set of Σ1 sentences which are absolutely independent relative to Peano arithmetic. The first proof uses the existence of an absolutely independent Π1 set of natural numbers, and the second uses a modification of the method of Gödel and Rosser.

Theories incomparable with respect to relative interpretability

1962 ◽  
Vol 27 (2) ◽  
pp. 195-211 ◽  
Author(s):  
Richard Montague
Keyword(s):  
Positive Integer ◽  
The Other ◽  
Natural Numbers ◽  
Infinite Set

The present paper concerns the relation of relative interpretability introduced in [8], and arises from a question posed by Tarski: are there two finitely axiomatizable subtheories of the arithmetic of natural numbers neither of which is relatively interpretable in the other? The question was answered affirmatively (without proof) in [3], and the answer was generalized in [4]: for any positive integer n, there exist n finitely axiomatizable subtheories of arithmetic such that no one of them is relatively interpretable in the union of the remainder. A further generalization was announced in [5] and is proved here: there is an infinite set of finitely axiomatizable subtheories of arithmetic such that no one of them is relatively interpretable in the union of the remainder. Several lemmas concerning the existence of self-referential and mutually referential formulas are given in Section 1, and will perhaps be of interest on their own account.

The classical and the ω-complete arithmetic

1958 ◽  
Vol 23 (2) ◽  
pp. 188-206 ◽  
Author(s):  
A. Grzegorczyk ◽  
A. Mostowski ◽  
C. Ryll-Nardzewski
Keyword(s):  
Positive Integer ◽  
Second Order ◽  
Natural Numbers ◽  
Formal Systems ◽  
Two Systems ◽  
Distinct Individual

We consider two formal systems for the theory of (natural) numbers, both of which are applied second-order functional calculi with equality and the description operator. The two systems have the same primitive symbols, rules of formation, and axioms, differing only in the rules of inference.The primitive logical symbols of the systems are the improper symbols (,), the prepositional connectives ∨, &, ⊃, ≡, ~, the quantifiers ( ), (E), the equality symbol =, the description operator ι,-infinitely many distinct individual (or number) variables, and for each positive integer k infinitely many distinct k-place function variables. Our systems have in addition the following four primitive nonlogical (or arithmetical) constants:0, 1, +, ×.The classes of “number formulas” (nfs) and “propositional formulas” (pfs) are defined inductively as the least classes of formal expressions (i.e. of concatenations of primitive symbols) satisfying the following conditions:(1) 0, 1, and the number variables are nfs.

scholarly journals Trace of Positive Integer Power of Squared Special Matrix

CAUCHY ◽  
2021 ◽  
Vol 6 (4) ◽  
pp. 200-211
Author(s):  
Rahmawati Rahmawati ◽  
Aryati Citra ◽  
Fitri Aryani ◽  
Corry Corazon Marzuki ◽  
Yuslenita Muda
Keyword(s):  
Positive Integer ◽  
Direct Proof ◽  
Integer Power ◽  
The Matrix ◽  
Special Matrix ◽  
Matrix Trace

The rectangle matrix to be discussed in this research is a special matrix where each entry in each line has the same value which is notated by An. The main aim of this paper is to find the general form of the matrix trace An powered positive integer m. To prove whether the general form of the matrix trace of An powered positive integer can be confirmed, mathematics induction and direct proof are used.  

scholarly journals Geometric Construction of Some Lehmer Means

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 251 ◽  
Author(s):  
Ralph Høibakk ◽  
Dag Lukkassen ◽  
Annette Meidell ◽  
Lars-Erik Persson
Keyword(s):  
Positive Integer ◽  
Geometric Construction ◽  
Geometric Constructions ◽  
Line Segments ◽  
Integer Power

The main aim of this paper is to contribute to the recently initiated research concerning geometric constructions of means, where the variables are appearing as line segments. The present study shows that all Lehmer means of two variables for integer power k and for k = m 2 , where m is an integer, can be geometrically constructed, that Lehmer means for power k = 0 , 1 and 2 can be geometrically constructed for any number of variables and that Lehmer means for power k = 1 / 2 and - 1 can be geometrically constructed, where the number of variables is n = 2 m and m is a positive integer.

Countable Compagtifications

1966 ◽  
Vol 18 ◽  
pp. 616-620 ◽  
Author(s):  
Kenneth D. Magill
Keyword(s):  
Positive Integer ◽  
Compact Space ◽  
Real Line ◽  
Topological Spaces ◽  
Dense Subspace ◽  
Natural Numbers ◽  
Finite Sets ◽  
The Real ◽  
One To One

It is assumed that all topological spaces discussed in this paper are Hausdorff. By a compactification αX of a space X we mean a compact space containing X as a dense subspace. If, for some positive integer n, αX — X consists of n points, we refer to αX as an n-point compactification of X, in which case we use the notation αn X. If αX — X is countable, we refer to αX as a countable compactification of X. In this paper, the statement that a set is countable means that its elements are in one-to-one correspondence with the natural numbers. In particular, finite sets are not regarded as being countable. Those spaces with n-point compactifications were characterized in (3). From the results obtained there it followed that the only n-point compactifications of the real line are the well-known 1- and 2-point compactifications and the only n-point compactification of the Euclidean N-space, EN (N > 1), is the 1-point compactification.

scholarly journals The Adjacency Matrix of One Type of Directed Graph and the Jacobsthal Numbers and Their Determinantal Representation

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Fatih Yılmaz ◽  
Durmuş Bozkurt
Keyword(s):  
Graph Theory ◽  
Positive Integer ◽  
Directed Graph ◽  
Adjacency Matrix ◽  
Intensive Study ◽  
Integer Power ◽  
Determinantal Representation ◽  
Adjacency Matrices

Recently there is huge interest in graph theory and intensive study on computing integer powers of matrices. In this paper, we consider one type of directed graph. Then we obtain a general form of the adjacency matrices of the graph. By using the well-known property which states the(i,j)entry ofAm(Ais adjacency matrix) is equal to the number of walks of lengthmfrom vertexito vertexj, we show that elements ofmth positive integer power of the adjacency matrix correspond to well-known Jacobsthal numbers. As a consequence, we give a Cassini-like formula for Jacobsthal numbers. We also give a matrix whose permanents are Jacobsthal numbers.

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