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[11] viXra:2105.0180 [pdf] replaced on 2021-07-29 21:32:14
Authors: Ryong Gil Choe
Comments: 16 pages, 2 tables
There have been published many research results on the Riemann hypothesis.
In this paper, we first find a new inequality for the Riemann hypothesis on the basis of well-known Robin theorem. Next, we introduce the error terms suitable to Mertens' formula and Chebyshev's function, and obtain their estimates. With such estimates and primorial numbers, we finally prove that the new inequality holds unconditionally.
Category: Number Theory
[10] viXra:2105.0171 [pdf] submitted on 2021-05-30 17:50:30
Authors: Marko V. Jankovic
Comments: 15 Pages.
In this paper an elementary proof of Green-Tao theorem is going to be presented. The proof represents an
extension of the proof of the Polignac's conjecture (or twin prime, or Sophie Germain primes conjecture). It will be
shown that arithmetic progressions that consist of prime numbers and that are of the length k (k is natural number),
could be obtained through k-stage recursive type sieve process, and that their number is infinite.
Category: Number Theory
[9] viXra:2105.0170 [pdf] replaced on 2021-06-15 08:47:45
Authors: Mar Detic
Comments: 3 Pages.
We introduce another way to enumerate primes up to N using 2x+1 and the summation of a constant. By which can also be used for primality test of a given integer.
Category: Number Theory
[8] viXra:2105.0124 [pdf] submitted on 2021-05-21 11:47:07
Authors: Salman Mahmud
Comments: 8 Pages.
Here we have shown a heuristic and approximate solution to the unsolved oppermann’s conjecture. Firstly
we have generated a formula to calculate the approximate number of multiples of a prime number ���� less than or
equal to a number �� which are not the multiples of the prime numbers ��1, ��2, ��3, … , ����−1 and ���� < ��. Then we have
generated another formula to calculate the number of prime numbers less than or equal to a number �� if the prime
numbers less than √�� are given where √�� ≤ �� ≤ �� . By using these formulas and the main concept of these formulas
we have presented our solution.
Category: Number Theory
[7] viXra:2105.0118 [pdf] replaced on 2024-03-12 23:41:54
Authors: Theophilus Agama
Comments: 9 Pages. This is a much simplified version of this paper with improved exposition.
This paper is an extension program of the notion of circle of partition developed in our first paper [1]. As an application we prove the Erdos-Turan additive base conjecture.
Category: Number Theory
[6] viXra:2105.0093 [pdf] replaced on 2025-10-14 20:34:16
Authors: Mar Detic
Comments: 5 Pages.
We present an algebraic and combinatorial formulation of Goldbach-type representations of integers as sums of primes. Every odd prime is written in the canonical linear form 2a + 1 (with a ∈ N0) and the sum of k primes becomes a linear Diophantine constraint on the corresponding a-variables. This transforms the problem into the study of integer lattice points on hyperplanes of the form Pk i=1 ai = N −k2 , together with primality filters on the linear forms 2ai + 1. We relate this combinatorial perspective to classical analytic heuristics (Hardy—Littlewood), additive-combinatorics sumset language, and partition/stars-and-bars counting, and we provide illustrative examples and an inline visualization for the k = 2 case.
Category: Number Theory
[5] viXra:2105.0087 [pdf] submitted on 2021-05-15 01:38:35
Authors: Yuji Masuda
Comments: 7 Pages.
By clearly defining the two concepts of ∞ and 0, we can prove various theorems and unsolved conjectures from our new knowledge about numbers. And not only that, the interpretation of this definition can be extended to physics.
Category: Number Theory
[4] viXra:2105.0073 [pdf] submitted on 2021-05-13 15:10:39
Authors: Dmitri Martila
Comments: 2 Pages. Submitted to the journal.
In this short note, I am disproving the Riemann Hypothesis.
Category: Number Theory
[3] viXra:2105.0046 [pdf] replaced on 2021-05-25 09:41:25
Authors: Juan Elias Millas Vera
Comments: 2 Pages.
This paper is a solution for the problem of optimization in the discomposing a composite natural number formed by the product of two primes.
Category: Number Theory
[2] viXra:2105.0012 [pdf] replaced on 2021-10-27 04:43:21
Authors: Jackson Lawrence Capper
Comments: 3 Pages.
We derive a generalised proof of Polignac’s Conjecture by regarding a prime in terms of its whole indivisibility and the consequent probability of prime intervals by applying the second Borel–Cantelli lemma. The specific case of the Twin Prime Conjecture is proven as an example.
Category: Number Theory
[1] viXra:2105.0003 [pdf] replaced on 2021-09-24 13:39:38
Authors: Leszek Mazurek
Comments: Pages.
In this paper, we prove the Collatz conjecture. The proof consists of two parts. The first, shows that the Collatz conjecture is the equivalent of the statement that every positive integer can be presented as a certain equation. In the second part, we prove that for every initial positive integer, this equation can be found. To achieve this, we propose a procedure that can be iterated, and we prove that by doing this we arrive at this equation. We also prove that any initial positive integer can be presented in an infinite number of ways in the form of needed equation. Each such form represents the loop occurring when number 1 is reached. The analysis is conducted using binary representation of numbers.
Category: Number Theory
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