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[26] viXra:2203.0183 [pdf] submitted on 2022-03-31 21:00:49
Authors: Michael Williams
Comments: 25 Pages. CC BY 4.0 license, DOI: 10.20944/preprints202203.0401.v1
Collatz conjecture (3n+1 problem) is an application of Cantor's isomorphism theorem (Cantor-Bernstein) under recursion. The set of 3n+1 for all odd positive integers n, is an order isomorphism for (odd X, 3X+1). The other (odd X, 3X+1) linear order has been discovered as a bijective order-embedding, with values congruent to powers of four. This is demonstrated using a binomial series as a set rule, then showing the isomorphic structure, mapping, and cardinality of those sets. Collatz conjecture is representative of an order machine for congruence to powers of two. If an initial value is not congruent to a power of two, then the iterative program operates the (odd X, 3X+1) order isomorphism until an embedded value is attained. Since this value is a power of four, repeated division by two tends the sequence to one. Because this same process occurs, regardless of the initial choice for a positive integer, Collatz conjecture is true.
Category: Number Theory
[25] viXra:2203.0170 [pdf] replaced on 2024-06-28 20:59:06
Authors: Theophilus Agama
Comments: 7 Pages.
In this paper we introduce and develop the notion of spanning of integers along functions $f:mathbb{N}longrightarrow mathbb{R}$. We apply this method to a class of problems requiring to determine if the equations of the form $tf(n)=n-k$ has a solution $nin mathbb{N}$ for a fixed $kin mathbb{N}$ and some $tin mathbb{N}$. In particular, we show that begin{align}# {nleq s~|~tvarphi(n)+1=n,~mathbf{for~some}~tin mathbb{N}}geq frac{s}{2log s}prod limits_{p | lfloor sfloor }(1-frac{1}{p})^{-1}-frac{3}{2}e^{gamma}onumberend{align}for $sgeq s_o$, where $varphi$ is the Euler totient function and $gamma=0.5772cdots$ is the Euler-Macheroni constant
Category: Number Theory
[24] viXra:2203.0169 [pdf] submitted on 2022-03-29 20:34:00
Authors: Marko V. Jankovic
Comments: 5 Pages.
In this paper, an addition tensor, or A-tensor is going to be presented. This is done by analogy to the recently introduced multiplication tensor or M-tensor. By comparing sub-tensors of A-tensor and M-tensor it is going to be proved that (strong) Goldbach's conjecture can not hold.
Category: Number Theory
[23] viXra:2203.0162 [pdf] submitted on 2022-03-28 15:49:14
Authors: Yuji Masuda
Comments: 4 Pages.
In this short paper, I will prove that matrices by my previous definition are interchangeable.
Category: Number Theory
[22] viXra:2203.0141 [pdf] submitted on 2022-03-24 09:59:39
Authors: Yuji Masuda
Comments: 3 Pages.
The purpose of this short paper is a matrix comparison by definition of the most recent previous short paper
Category: Number Theory
[21] viXra:2203.0129 [pdf] replaced on 2022-04-26 07:09:05
Authors: Hajime Mashima
Comments: 22 Pages.
The more difficult the problem, the more limited the path.
Category: Number Theory
[20] viXra:2203.0128 [pdf] submitted on 2022-03-22 21:19:37
Authors: Ricardo G. Barca
Comments: 32 Pages. The manuscript was edited for proper English language by one editor at American Journal Experts (Certificate Verification Key: C0C3-5251-4504-E14D-BE84). However, afterwards some changes have been made.
The binary Goldbach conjecture asserts that every even integer greater than $4$ is the sum of two primes. In this paper, we prove that there exists an integer $K_\alpha > 4$ such that every even integer $x > p_k^2$ can be expressed as the sum of two primes, where $p_k$ is the $k$th prime number and $k > K_\alpha$. To prove this statement, we begin by introducing a type of double sieve of Eratosthenes as follows. Given a positive even integer $x > 4$, we sift from $[1, x]$ all those elements that are congruents to $0$ modulo $p$ or congruents to $x$ modulo $p$, where $p$ is a prime less than $\sqrt{x}$. Therefore, any integer in the interval $[\sqrt{x}, x]$ that remains unsifted is a prime $q$ for which either $x-q = 1$ or $x-q$ is also a prime. Then, we introduce a new way of formulating a sieve, which we call the sequence of $k$-tuples of remainders. By means of this tool, we prove that there exists an integer $K_\alpha > 4$ such that $p_k / 2$ is a lower bound for the sifting function of this sieve, for every even number $x$ that satisfies $p_k^2 < x < p_{k+1}^2$, where $k > K_\alpha$, which implies that $x > p_k^2 \; (k > K_\alpha)$ can be expressed as the sum of two primes.
Category: Number Theory
[19] viXra:2203.0122 [pdf] submitted on 2022-03-22 16:08:09
Authors: Yuji Masuda
Comments: 2 Pages.
First, ±∞ is constant at any observation point. If a set of real numbers is R, then On the other hand, when x (∈R)is taken on a number line, the absolute value X becomes larger toward ± ∞ as the absolute value X is expanded. Similarly, as the size decreases, the absolute value X decreases toward 0. Furthermore, x (-1) represents the reversal of the direction of the axis.
Category: Number Theory
[18] viXra:2203.0117 [pdf] submitted on 2022-03-21 09:27:06
Authors: Denis Gallet
Comments: 7 Pages.
I study several papers of V.S. Adamchik and I find several mistakes about integrals and the Melzak's product.In the same time, I give more general formulas of three integrals.
Category: Number Theory
[17] viXra:2203.0114 [pdf] submitted on 2022-03-21 12:18:40
Authors: Denis Gallet
Comments: 7 Pages.
This is the continuity of my precedent paper and I give six general formulas of six integrals.
Category: Number Theory
[16] viXra:2203.0104 [pdf] submitted on 2022-03-19 13:33:58
Authors: Yuji Masuda
Comments: 1 Page.
The purpose of this short paper is to prove that multiplying by a minus signifies a reverse rotation.
Category: Number Theory
[15] viXra:2203.0092 [pdf] submitted on 2022-03-17 22:03:37
Authors: Theophilus Agama
Comments: 3 Pages.
In this paper we prove an inequality relating the length of addition chains producing number of the form $2^n-1$ to the length of their shortest addition chain producing their exponents. In particular, we obtain the inequality $$\delta(2^n-1)\leq n-1+\iota(n)+G(n)$$ where $\delta(n)$ and $\iota(n)$ denotes the length of an addition chain and the shortest addition chain producing $n$, respectively, with $G:\mathbb{N}\longrightarrow \mathbb{R}$.
Category: Number Theory
[14] viXra:2203.0086 [pdf] submitted on 2022-03-15 07:48:09
Authors: José Alcauza
Comments: 5 Pages.
In this paper, we find a curious and simple possible solution to the critical line of nontrivial zeros
of Riemann zeta function.
Category: Number Theory
[13] viXra:2203.0084 [pdf] replaced on 2022-05-31 20:40:43
Authors: Wing K. Yu
Comments: 5 Pages.
In this paper, we will prove that when an integer n ˃1, there exists a prime number between 3n and 4n. This is another step in the expansion of the Bertrand’s postulate - Chebyshev’s theorem after the proof of a prime number between 2n and 3n.
Category: Number Theory
[12] viXra:2203.0076 [pdf] submitted on 2022-03-14 10:07:51
Authors: Yuji Masuda
Comments: 1 Page.
The objective was to prove the ABC conjecture by my definition series.
Category: Number Theory
[11] viXra:2203.0068 [pdf] submitted on 2022-03-13 07:38:48
Authors: James Edwin Rock
Comments: 2 Pages.
We examine the properties of a partition of the positive integers and exhibit some formulas that describe the partition and place all positive integers in specific sequences that comprise the partition. The Collatz conjecture also partitions the positive integers.
Category: Number Theory
[10] viXra:2203.0058 [pdf] submitted on 2022-03-12 05:01:33
Authors: Yuji Masuda
Comments: 1 Page.
{1, 2, 4, 5, 7, 8,...} has some interesting properties.
In particular, I will discuss their relationship to prime numbers.
Category: Number Theory
[9] viXra:2203.0057 [pdf] submitted on 2022-03-12 10:31:31
Authors: Arthur V. Shevenyonov
Comments: 5 Pages. multilayer duality
The ‘gray area,’ or indeed overlap, in between primes versus composites pertains to facets spanning beyond their belonging on the natural axis. The former can, for one, best be posited as building blocks with reference to a generalized rho-operation. At this rate, primes build on multiplicity (rho specified anywhere near 0) whereas composites on additivity (rho taken to 1). These result from symmetry (self-duality) versus averaging over the latter’s solutions, respectively, while pointing to scenarios such as “either/both” versus “neither/both” (suggesting the number of (#, X) prime materializations/hits for the prime versus [higher] composite subdomains). The “both” overlap (boasting a relatively lower proportion in the prime subdomain) may plausibly amount to the aforementioned inconclusive-necessity area. The prime power (i.e. the number of prime radicals in the putative composite, degenerate/singular not least) of (#, X) tends to that of the input prime/composite in question. Among other regularities, recurring sequences and collated generalizations are observed in a set of follow-on conjectures.
Category: Number Theory
[8] viXra:2203.0054 [pdf] submitted on 2022-03-11 03:27:51
Authors: Arthur V. Shevenyonov
Comments: 2 Pages. one more strand/straddle
It can and has been shown copiously that the nature of prime numbers could be viewed as recursive, Diophantine, self-spawned. Incidentally, it proves even simpler than that: Per any number prime, so likewise is either the sum or the difference of its ‘tweak’ characteristics.
Category: Number Theory
[7] viXra:2203.0050 [pdf] replaced on 2022-03-27 05:08:28
Authors: Timothy W. Jones
Comments: 6 Pages. Further simplifications of last proof.
We show that using the denominators of the terms of Zeta(n)-1=z_n as decimal bases gives all rational numbers in (0,1) as single decimals. We also show the partial sums of z_n are not given by such single digits using the partial sum's terms. These two properties yield a proof that z_n is irrational.
Category: Number Theory
[6] viXra:2203.0033 [pdf] replaced on 2022-04-03 07:45:48
Authors: Marcin Barylski
Comments: 5 Pages.
Based on primality property change for two consecutive terms of infinite sequence of natural numbers it is possible to establish rules which allow to draw a figure. This work is devoted to studies on various sequences of natural numbers which produced interesting 2D and 3D outputs.
Category: Number Theory
[5] viXra:2203.0026 [pdf] replaced on 2022-03-17 13:52:34
Authors: P. Aaron Bloom
Comments: 2 Pages.
We parametrize the Fermat equation, X ^ n + Y ^ n = Z ^ n, with a form that explicitly shows no integral solution for odd n > 1.
Category: Number Theory
[4] viXra:2203.0024 [pdf] submitted on 2022-03-04 03:40:56
Authors: Arthur Shevenyonov
Comments: 13 Pages. standard results in the lit could be inferred more efficiently, too
All of the key Conjectures that may still be of interest beyond, or irrespective of, assuming the validity of RH appear straightforward to tackle with a handful of minimalist instruments based on the RI solely. A host of extra Propositions (zeta based formulae & equivalences) being spawned from the latter pillar as a ‘side effect’ of importance in its own right.
Category: Number Theory
[3] viXra:2203.0021 [pdf] submitted on 2022-03-04 21:31:14
Authors: Seiji Tomita, Oliver Couto
Comments: 12 Pages.
We consider two types of equations shown below:
ax^4+ by^4= cz^4+ dw^4
ax^4+ by^4+ cz^4= dw^4
Condition: product (abcd) not equal to zero
Existence of solution for Diophantine equation:
ax^4+ by^4= cz^4+ dw^4 & ax^4+ by^4+ cz^4= dw^4, are known if (abcd) is square number& product not equal to zero. So we are curious about whether above equation has a solution if (abcd) is not square number & product not equal to zero. In particular, when does this equation have infinitely many integer solutions? Bremner, A., & Choudhry, A., & Ulas [1] have showed the solution family of the similar equation ( ax^4+ by^4+ cz^4+ dw^4= 0 ) with infinitely many rational points using elliptic curve theory. We show other family of solution of this equation with infinitely many rational points. As a bonus we have considered an interesting case of equation (ax^4+ by^4= az^4+ bw^4) in which the product of the coefficents is a square number, but has been parameterized by using only algebraic methods without taking re-course to elliptic curve theory.
Category: Number Theory
[2] viXra:2203.0013 [pdf] submitted on 2022-03-02 18:55:45
Authors: Zhong Wang
Comments: 6 Pages.
Antique prime number problems are generalized as one family and transformed into quantitative domination.
Introducing prime pair, the longest gap in Moiré Pattern, remainder sequence, primorial ring and etc.
A makeover view of number as a closed ring.
Category: Number Theory
[1] viXra:2203.0007 [pdf] submitted on 2022-03-01 03:03:06
Authors: E. Zhou, M. Zhou
Comments: 2 Pages.
There are the closed-form representations of the Euler’s Zeta function at odd positive integers in this article.
Category: Number Theory
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