Odd Natural Number Research Articles

Ramanujan Sums of All Natural Numbers with Grandi Series

We are all know that the Ramanujan’s theory of calculating the sum of all the natural number. When we came to known that sum of all-natural number is ,that time everyone thinks about two things, one is how it possible that sum of positive number is negative & other is how it is very close to zero. Another research paper gives value of all natural number is then I started the study of this sum, then I find it is as zero. Quit interestingly but this value has more important than Ramanujan’s sum because I did not consider the Grandi series for calculating this sum. And then it is very easy to calculate the sum of negative integer & sum of all odd natural number & sum of all even natural number.

A Study On Maximal Embedding Dimension Numerical Semigroups

In this article, it is examine some the numerical semigroups W and W/2 such that W= < p, q > and W/2= < p, p+q/2, q > where p < q and p, q are odd natural numbers.

Characteristic polynomial of maximum and minimum matrix of square power graph of dihedral group of order 2n with odd natural number n

Characteristic polynomial of maximum and minimum matrix of square power graph of dihedral group of order 2n with odd natural number n

Characterization of prime and composite numbers using the notion of successive sum of integers and the consequence in primality testing

In this paper, we give a characterization of primes and composite natural numbers using the notion of the sum of successive natural numbers. We prove essentially that an odd natural number N ≥ 3 is prime if and only if the unique decomposition of N as a sum of successive natural numbers is the trivial decomposition N = a + (a + 1) with a = (N - 1) / 2.

Non-Imaginary unit Circle and Distribution Odd Natural Numbers

This paper introduces a non-Imaginary unit circle partitioning as proof for the distribution of odd natural numbers in relation to an imaginary unit circle in a complex plane. First, we will introduce the concept of a non-imaginary unit circle and its relation to an imaginary unit circle in a complex plane. Then we will go through some examples to prove that for any N odd natural number at N/2, we only have the imaginary part for any complex number on the complex plane if we use our technique of portioning for the non-imaginary unit circle.

The Riemann Hypothesis Applies Not Only to Prime Numbers

The analysis of a series of natural numbers 0, 1, 2, 3, 4, ..., 1024 is given after expansion in the binary number system. Particular cases of this series are composite, even and odd numbers, and the latter are divided into prime numbers and composite odd numbers. Comparison of these series is carried out according to the general formula for the distribution of binary numbers for all 9 digits of the binary number system. It is shown that the shift of the oscillation of a binary number is different for varieties of natural numbers. It is proved that the real root of the Riemann zeta function 1/2 exists for any series of numbers obtained from the series of natural numbers. In the limit, with an increase in the power of the series, a sinusoid is subtracted from the real root (for series of odd, prime and odd composite numbers) and a cosine function (for series of natural, composite and integer numbers), the amplitude of which is also equal to 1/2, and the half-period of the trigonometric function are two numbers: 1 —for series of natural and composite natural numbers; 2—for series of odd natural, prime, composite odd and even numbers. Moreover, under the sine and cosine functions, the varieties of series of natural numbers are located on the Riemannian critical lines in the following way: a) on the zero vertical of the binary expansion of series of natural and composite numbers, the value π; b) on the first vertical of the binary expansion of series of odd, prime, composite odd and integer natural numbers π/2.

Prime Representing Polynomial with 10 Unknowns

Summary In this article we formalize in Mizar [1], [2] the final step of our attempt to formally construct a prime representing polynomial with 10 variables proposed by Yuri Matiyasevich in [4]. The first part of the article includes many auxiliary lemmas related to multivariate polynomials. We start from the properties of monomials, among them their evaluation as well as the power function on polynomials to define the substitution for multivariate polynomials. For simplicity, we assume that a polynomial and substituted ones as i-th variable have the same number of variables. Then we study the number of variables that are used in given multivariate polynomials. By the used variable we mean a variable that is raised at least once to a non-zero power. We consider both adding unused variables and eliminating them. The second part of the paper deals with the construction of the polynomial proposed by Yuri Matiyasevich. First, we introduce a diophantine polynomial over 4 variables that has roots in integers if and only if indicated variable is the square of a natural number, and another two is the square of an odd natural number. We modify the polynomial by adding two variables in such a way that the root additionally requires the divisibility of these added variables. Then we modify again the polynomial by adding two variables to also guarantee the nonnegativity condition of one of these variables. Finally, we combine the prime diophantine representation proved in [7] with the obtained polynomial constructing a prime representing polynomial with 10 variables. This work has been partially presented in [8] with the obtained polynomial constructing a prime representing polynomial with 10 variables in Theorem (85).

Analyzing the Collatz Conjecture Using the Mathematical Complete Induction Method

In this paper, we demonstrate the Collatz conjecture using the mathematical complete induction method. We show that this conjecture is satisfied for the first values of natural numbers, and in analyzing the sequence generated by odd numbers, we can deduce a formula for the general term of the Collatz sequence for any odd natural number n after several iterations. This formula is used in one case that we analyze using the mathematical complete induction method in the process of demonstrating the conjecture.

Chebyshev polynomials and Galois groups of De Moivre polynomials

Let [Formula: see text] be an odd natural number. In 1738, Abraham de Moivre introduced a family of polynomials of degree n with rational coefficients, all of which are solvable. So far, the Galois groups of these polynomials have been investigated only for prime numbers [Formula: see text] and under special assumptions. We describe the Galois groups for arbitrary odd numbers [Formula: see text] in the irreducible case, up to few exceptions. In addition, we express all zeros of such a polynomial as rational functions of three zeros, two of which are connected in a certain sense. These results are based on the reduction of an irrational of degree 2n to irrationals of degree [Formula: see text]. Such a reduction was given in a previous paper of the author. Here, however, we present a much simpler approach that is based on properties of Chebyshev polynomials. We also give a simple proof of a result of Filaseta et al.

Some hyperstability results of a p-radical functional equation related to quartic mappings in non-archimedean Banach spaces

The aim of this paper is to introduce and solve the following p-radical functional equation related to quartic mappings where f is a mapping from R into a vector space X and p ≥ 3 is an odd natural number. Using an analogue version of Brzd¸ek’s fixed point theorem [13], we establish some hyperstability results for the considered equation in non-Archimedean Banach spaces. Also, we give some hyperstability results for the inhomogeneous p-radical functional equation related to quartic mapping.

Some hyperstability results of a p-radical functional equation related to Drygas mappings in non-Archimedean Banach spaces

The aim of this paper is to introduce and solve the following pradical functional equation related to Drygas mappings f(√(p&x^p+ y^p ))+f(√(p&x^p+ y^p ))=2f(x)+f(y)+f(-y),x,y ∈R, where f is a mapping from R into a vector space X and p ≥ 3 is an odd natural number. Using an analogue version of Brzdȩk’s fixed point theorem [14], we establish some hyperstability results for the considered equation in non-Archimedean Banach spaces. Also, we give some hyperstability results for the inhomogeneous p-radical functional equation related to Drygas mappings f(√(p&x^p+ y^p ))+f(√(p&x^p+ y^p ))=2f(x)+f(y)+f(-y)+G(x,y)

Computations of the least number of periodic points of smooth boundary-preserving self-maps of simply-connected manifolds

Let $r$ be an odd natural number, $M$ a compact simply-connected smooth manifold, $\dim M\geq 4$, such that its boundary $\partial M$ is also simply-connected. We consider $f$, a $C^1$ self-maps of $M$, preserving $\partial M$. In [G. Graff and J. Jezierski, Geom. Dedicata 187 (2017), 241-258] the smooth Nielsen type periodic number $D_r(f;M,\partial M)$ was defined and proved to be equal to the minimal number of $r$-periodic points for all maps preserving $\partial M$ and $C^1$-homotopic to $f$. In this paper we demonstrate a purely combinatorial method of calculation of the invariant and illustrate it in various cases.

Fixed Point Sets of k-Continuous Self-Maps of m-Iterated Digital Wedges

Let Ckn,l be a simple closed k-curves with l elements in Zn and W:=Ckn,l∨⋯∨Ckn,l︷m-times be an m-iterated digital wedges of Ckn,l, and F(Conk(W)) be an alignment of fixed point sets of W. Then, the aim of the paper is devoted to investigating various properties of F(Conk(W)). Furthermore, when proceeding with this work, this paper addresses several unsolved problems. To be specific, we firstly formulate an alignment of fixed point sets of Ckn,l, denoted by F(Conk(Ckn,l)), where l(≥7) is an odd natural number and k≠2n. Secondly, given a digital image (X,k) with X♯=n, we find a certain condition that supports n−1,n−2∈F(Conk(X)). Thirdly, after finding some features of F(Conk(W)), we develop a method of making F(Conk(W)) perfect according to the (even or odd) number l of Ckn,l. Finally, we prove that the perfectness of F(Conk(W)) is equivalent to that of F(Conk(Ckn,l)). This can play an important role in studying fixed point theory and digital curve theory. This paper only deals with k-connected digital images (X,k) such that X♯≥2.

Reducing radicals in the spirit of Euclid

Let p be an odd natural number ≥3. Inspired by results from Euclid's Elements, we express the irrationaly=d+Rp, whose degree is 2p, as a polynomial function of irrationals of degrees ≤p. In certain cases y is expressed by simple radicals. This reduction of the degree exhibits remarkably regular patterns of the polynomials involved. The proof is based on hypergeometric summation, in particular, on Zeilberger's algorithm.

Collatz conjecture revisited: an elementary generalization

Abstract Collatz conjecture states that iterating the map that takes even natural number n to n 2 {n \over 2} and odd natural number n to 3n + 1, will eventually obtain 1. In this paper a new generalization of the Collatz conjecture is analyzed and some interesting results are obtained. Since Collatz conjecture can be seen as a particular case of the generalization introduced in this articule, several more general conjectures are also presented.

Shimura and Shintani liftings of certain cusp forms of half-integral and integral weights

In [4], W. Kohnen constructed explicit Shintani lifts from the space of cusp forms of weight $2k$ on $Γ_0(N)$, which are mapped into the plus space $S^+_{k+1/2}(Γ_0(4N))$, where $N$ is an odd integer. This lifting is adjoint to the (modified) Shimura map defined by him with respect to the Petersson scalar product. Using this construction along with the theory of newforms on $S^+_{k+1/2}(Γ_0(4N))$ (where $N$ is an odd square-free natural number) developed in [3], he derived explicit Waldspurger theorem for the newforms belonging to the space $S^+_{k+1/2}(Γ_0(4N))$. Further in this work, Kohnen considered the $\pm 1$ eigen subspaces corresponding to the Atkin-Lehner involution and showed that the intersection of this $\pm$ spaces with the corresponding newform spaces are isomorphic under the Shimura correspondence. A natural question is whether a similar result can be obtained for the intersection of this $(\pm1)$ subspaces with the space of oldforms. In this direction, S. Choi and C. H. Kim [2] considered the case where $N$ is an odd prime $p$ and constructed similar Shimura and Shintani maps between subspaces of forms of half-integral and integral weights. The subspaces considered by Choi and Kim are nothing but $+1$ eigenspace under the Fricke involution. In this paper, we generalise the work of Choi and Kim to the case where $N$ is an odd square-free natural number.

Generalization of Erdős-Kac theorem

Let ω(n) is the number of distinct prime factors of the natural number n; we consider two cases where l is even and odd natural numbers, and then we prove a more general form of the classical Erdős-Kac theorem.

On a recent reciprocity formula for Dedekind sums

Let [Formula: see text] denote the classical Dedekind sum and [Formula: see text]. Recently, Du and Zhang proved the following reciprocity formula. If [Formula: see text] and [Formula: see text] are odd natural numbers, [Formula: see text], then [Formula: see text] where [Formula: see text] and [Formula: see text]. In this paper, we show that this formula is a special case of a series of similar reciprocity formulas. Whereas Du and Zhang worked with the connection of Dedekind sums and values of [Formula: see text]-series, our main tool is the three-term relation for Dedekind sums.

On braided almost square-free fusion categories with odd dimension

Let q be a odd prime number, d be an odd square-free natural number. We prove that a braided fusion category of Frobenius–Perron dimension dqn is solvable. If then we further prove that is tensor equivalent to the category of finite-dimensional representations of a semisimple quasitriangular Hopf algebra which is kG for some abelian group G, or a crossed product for some σ.

Asymptotic formula for sum of moment mean deviation for order statistics from uniform distribution

Assume that [Formula: see text] mobile sensors are thrown uniformly and independently at random with the uniform distribution on the unit interval. We study the expected sum over all sensors [Formula: see text] from [Formula: see text] to [Formula: see text] where the contribution of the [Formula: see text] sensor is its displacement from the current location to the anchor equidistant point [Formula: see text] raised to the [Formula: see text] power, when [Formula: see text] is an odd natural number. As a consequence, we derive the following asymptotic identity. Fix [Formula: see text] positive integer. Let [Formula: see text] denote the [Formula: see text] order statistic from a random sample of size [Formula: see text] from the Uniform[Formula: see text] population. Then [Formula: see text] where [Formula: see text] is the Gamma function.