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[16] viXra:2104.0172 [pdf] submitted on 2021-04-26 21:35:58
Authors: Takamasa Noguchi
Comments: 10 Pages.
There are already various formulas for calculating power remainders and roots of remainders.
Based on these, I have created a simple and quick way to calculate it.
However, there is no theoretical proof.
Category: Number Theory
[15] viXra:2104.0165 [pdf] submitted on 2021-04-27 17:52:27
Authors: Suaib Lateef
Comments: 4 pages
In this paper, we prove that the sum of fifth powers of the first n natural numbers is a difference of two squares whose difference is the sum of the natural numbers. We also conjecture that the same is true for all odd powers.
Category: Number Theory
[14] viXra:2104.0153 [pdf] submitted on 2021-04-25 08:02:03
Authors: O. Kurwa
Comments: 1 Page.
A long standing conjecture by Goldbach states that every even integer greater than two can be written as a sum of two prime numbers. In this paper, we present the proof that the conjecture is true for 60.
Category: Number Theory
[13] viXra:2104.0138 [pdf] replaced on 2021-05-19 18:27:16
Authors: Dante Servi
Comments: 10 Pages. Copyright by Servi Dante.
My research began with the distribution of primes and composite numbers, I believe I have identified the mechanism. In this article I deal with the factorization of composite numbers or the search for prime numbers that multiplied together give the composite number under examination as a result. From the very beginning of the research I came across those sequences of prime numbers and composite numbers that I called combined sequences, continuing the research I believe I have revealed all the main characteristics. In this article I propose a table that derives strictly from the combined sequence that I had already called [Sc<=11], for this reason I have identified it with the same name.
Category: Number Theory
[12] viXra:2104.0098 [pdf] replaced on 2021-07-03 06:17:21
Authors: A. A. Frempong
Comments: 6 Pages. Copyright © by A. A. Frempong
The symmetric structure of the Beal conjecture proof, in this paper, exemplifies the beauty in mathematics. The author applies basic mathematical principles to surely, instructionally, and beautifully, prove the original Beal conjecture that if A^x + B^y = C^z, where A, B, C, x, y, z are positive integers and x, y, z > 2, then A, B and C have a common prime factor. One will let r, s, and t be prime factors of A, B and C, respectively, such that A = Dr, B = Es, C = Ft, where D, E, and F are positive integers. Then, the equation A^x + B^y = C^z becomes D^xr^x + E^ys^y = F^zt^z. The proof would be complete after showing that the equalities, r^x = t^x, s^y = t^y and r = s = t, are true. The proof of the above equalities would involve showing that the ratios, (r^x)/(t^x) = 1 and (s^y)/(t^y) =1, which would imply that r = s = t. The main principle for obtaining relationships between the prime factors on the left side of the equation and the prime factor on the right side of the equation is that the power of each prime factor on the left side of the equation equals the same power of the prime factor on the right side of the equation. High school students can learn and prove this conjecture for a bonus question on a final class exam.
Category: Number Theory
[11] viXra:2104.0088 [pdf] replaced on 2021-05-05 17:54:27
Authors: N. Scott
Comments: 5 Pages. [Corrections made by viXra Admin to conform with the rules of viXra.org]
In this paper I will be proving that Re(z) being equal to more than one is the convergent half-plane beyond s>1. That of which is the pole or singularity of the whole functional system. I will be providing a counter-example and a forth-wright approach to the Riemann Hypothesis, Riemann Zeta Function. In the beginning I assumed that the calculations from these unreliable third-party sources of calculation were just normal. But then I was able to finally crack the problem of inserting the Riemann Zeta Function into an image of the formula.
Category: Number Theory
[10] viXra:2104.0083 [pdf] replaced on 2021-04-30 10:24:29
Authors: Dante Servi
Comments: 6 Pages. Copyright by Servi Dante.
The mechanism that generates prime numbers and composite numbers requires an infinite series of cycles, each of which takes place in two pass; at the end of each cycle all the information possible and necessary to continue is obtained.
I made an image that illustrates the first four cycles starting from 1; I maintain that although it is a mechanism that acts on numbers, with only four cycles of the graphic method I have adopted, I provide the demonstration of the mechanism without using any calculation.
Category: Number Theory
[9] viXra:2104.0080 [pdf] replaced on 2021-06-07 12:52:16
Authors: Ryan Zielinski
Comments: 7 Pages. This work is licensed under the CC BY 4.0, a Creative Commons Attribution License.
Let "Faulhaber's formula" refer to an expression for the sum of powers of integers written with terms of n(n+1)/2. Initially, the author used Faulhaber's formula to explain why odd Bernoulli numbers are equal to zero. Next, Cereceda gave alternate proofs of that result and then proved the converse, if odd Bernoulli numbers are equal to zero then we can derive Faulhaber's formula. Here, the original author will give a new proof of the converse.
Category: Number Theory
[8] viXra:2104.0079 [pdf] submitted on 2021-04-12 21:37:04
Authors: Nikolaus Castell-Castell, Tom Hermann Tietken
Comments: 13 Pages. [Corrections made by viXra Admin to conform with the requirements on the Submission Form]
In a certain part of the logic (where many identical individual parts are not changed or destroyed, but only rearranged), a result that comes about using a simple basic calculation should also be able to be traced back using a simple basic calculation. If it is easy to get a certain result when you multiply two numbers (factors) together, it should not be difficult to "recalculate" this result back to its two factors with an equally simple basic calculation. This present work explicitly does not deal with the above-mentioned procedure, how a number, which has arisen from the multiplication of two numbers (in the practice of cryptography, these are prime numbers), is again broken down into these two prime numbers ("factored") ! The present manuscript is only intended to illustrate what is possible with the application of the basic calculation types, which are accessible to everyone. For this purpose we have asked ourselves how one can even without an algorithm that directly breaks down (factored) "large numbers" (products), can come to similarly useful results. A suggested solution could be: You can also indirectly identify the prime numbers from which large products have arisen by creating lists in which all the numbers in question are stored in an orderly manner, continuously increasing.
Category: Number Theory
[7] viXra:2104.0076 [pdf] submitted on 2021-04-12 03:50:37
Authors: Theophilus Agama
Comments: 9 Pages.
Motivated by Gilbreath's conjecture, we develop the notion of the gap sequence induced by any sequence of numbers. We introduce the notion of the path and associated circuits induced by an originator and study the conjecture via the notion of the trace and length of a path.
Category: Number Theory
[6] viXra:2104.0072 [pdf] replaced on 2021-10-16 08:55:54
Authors: Timothy W. Jones
Comments: 23 Pages. Improvements
We develop definitions and a theory for convergent series that have terms of the form $1/a_j$ where $a_j$ is an integer greater than one and the series convergence point is less than one. These series have terms with denominators that can be used as number bases. The series for $e-2$ and $z_n=\zeta(n)-1$ are of this type. Further, both series yield number bases that can represent all possible rational convergence points as single digits. As partials for these series are rational numbers, all partials can be given as single decimals using some $a_j$ as a base. In the case of $e-2$, the last term of a partial yields such a base and partials form systems of nesting inequalities yielding a proof of the irrationality of $e-2$. Using limits in an unusual way we are able to give a second proof for the irrationality of $e-2$. A third proof validates the second using Dedekind cuts. In the case of $z_n$, using the $z_2$ case we determine that such systems of nesting inequalities are not formed, but we discover partials require bases greater than the denominator of their last term. We prove this property for the general $z_n$ case and, using the unusual limit style proof mentioned, prove $z_n$ is irrational. We once again validate the proof using Dedekind cuts. Finally, we are able to give what we consider a satisfying proof showing why both $e-2$ and $z_n$ are irrational.
Category: Number Theory
[5] viXra:2104.0062 [pdf] submitted on 2021-04-11 07:12:41
Authors: Timothy W. Jones
Comments: 14 Pages.
Although Beukers's proof that Zeta(2) and Zeta(3) are irrational are at the level of advanced calculus, they are condensed. This article slows down the development and adds examples of the techniques used. In so doing it is hoped that more people might enjoy these mathematical results. We focus on the easier of the two Zeta(2).
Category: Number Theory
[4] viXra:2104.0042 [pdf] submitted on 2021-04-08 22:06:02
Authors: T. Nakashima
Comments: 2 Pages.
For mobius function, there is some strange reration. I intend to test one theorem.
Category: Number Theory
[3] viXra:2104.0033 [pdf] submitted on 2021-04-07 12:18:19
Authors: Jesus Sanchez
Comments: 37 Pages.
In this paper, it will be shown that it is possible to detect if a number p is prime [1] or not, using the following expression involving the q-polygama function [2] (represented as ψ) and being c whatever positive real number higher than zero:
q_2=1/2π ∫_(-π)^π〖e^pjω ((ψ_q (1)+ln(1-e^(-c-jω) ))/(ln(e^(-c-jω) )) -(e^(-c-jω)+e^(-2c-2jω))/(1-e^(-c-jω) )) 〗 dω (1)
If the result is zero, p is a prime number. If it is different from zero, the result will give information about the number of factors that the number p has.
We will check that using this as a basis, we can obtain the following integral:
q_3=1/2π ∫_(-π)^π〖e^pjω (∑_(k=2)^(k=∞)〖e^(-c(2+k)) e^(-2kjω) 1/(1-〖e^(-c) e〗^(-kjω) )〗) 〗 dω (2)
And it is possible to obtain the sum of the factors of a semiprime number p [3] with it. Once we have the sum and the product of the factors, it is immediate to obtain the two factors of the semiprime number. The solution of that integral (solving it numerically) is obtained in polynomial time (quadratic).
To do so, the second element of the product inside the integral has had to be calculated before and stored in a table (data base). Once this prework is done, the result is given in polynomial time (linear) independently of p or its size. You can check that this is possible because the second element of the product within the integral does not depend on p:
∑_(k=2)^(k=∞)〖e^(-c(2+k)) e^(-2kjω) 1/(1-〖e^(-c) e〗^(-kjω) )〗 (3)
Category: Number Theory
[2] viXra:2104.0020 [pdf] replaced on 2021-04-18 21:27:01
Authors: John Yuk Ching Ting
Comments: 50 Pages. Rigorous Proofs on Riemann Hypothesis, Polignac's and Twin Prime Conjectures with incorporation of COVID-19.
COVID-19 originated from Wuhan, China in December 2019. Declared by the World Health Organization on March 11, 2020; COVID-19 pandemic has resulted in unprecedented negative global impacts on health and economy. International cooperation is required to combat this "Incompletely Predictable" pandemic. With manifestations of Chaos-Fractal phenomena, we mathematically model COVID-19 and solve [unconnected] open problems in Number theory using our versatile Fic-Fac Ratio. Computed as Information-based complexity, our innovative Information-complexity conservation constitutes a unique all-purpose analytic tool associated with Mathematics for Incompletely Predictable problems. These problems are literally "complex systems" containing well-defined Incompletely Predictable entities such as nontrivial zeros and two types of Gram points in Riemann zeta function (or its proxy Dirichlet eta function) together with prime and composite numbers from Sieve of Eratosthenes. Correct and complete mathematical arguments for first key step of converting this function into its continuous format version, and second key step of using our unique Dimension (2x - N) system instead of this Sieve result in primary spin-offs from first key step consisting of providing proof for Riemann hypothesis (and explaining closely related two types of Gram points), and second key step consisting of providing proofs for Polignac's and Twin prime conjectures.
Category: Number Theory
[1] viXra:2104.0002 [pdf] replaced on 2021-08-31 18:10:43
Authors: Federico Gabriel
Comments: 2 Pages.
This article proposes a new simply proof of Fermat's Last Theorem with Pythagorean triangles.
Category: Number Theory
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