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[25] viXra:2311.0139 [pdf] submitted on 2023-11-27 21:46:55
Authors: Ait saadi Ahcene
Comments: 4 Pages. (Correction made by viXra Admin to conform with scholarly norm)
In this article, I solve the general equation of degree 5 of the a particular form. For this I used Mathematics that I invented. The method I invented allows me to solve all equations of degrees greater than 4, as well as equations of another form: The principle, is to find for all these equations, an equation of degree three (3) which has at least one solution in common with those of degree greater than 4. With my method of course this is possible.
Category: Number Theory
[24] viXra:2311.0137 [pdf] replaced on 2023-12-03 18:59:14
Authors: Juan Moreno Borrallo
Comments: 6 Pages.
In this brief paper we study and bound Mertens function. The main breakthrough is the obtention of a Möbius-invertible formulation of Mertens function, which with some transformations and the application of a generalization of Möbius inversion formula, allows us to reach an asymptotic rate of growth of Mertens function that proves the Riemann Hypothesis.
Category: Number Theory
[23] viXra:2311.0126 [pdf] replaced on 2023-11-29 06:38:42
Authors: Isaac Mor
Comments: 13 Pages.
I came across "The Ramanujan Machine" on the Internet and, using my intuition on those kind of stuff, I found some interesting results.
Category: Number Theory
[22] viXra:2311.0125 [pdf] replaced on 2024-03-07 06:20:33
Authors: Tsuneaki Takahashi
Comments: 2 Pages.
If the series of Collatz Conjecture integer has looping in it, it is sure the members of the loop cannot reach to value 1. Here it is proven that the possibility of looping is zero except one.
Category: Number Theory
[21] viXra:2311.0119 [pdf] submitted on 2023-11-24 23:54:05
Authors: Leonardo de Lima
Comments: 8 Pages.
This article delves into the properties of the Riemann zeta function, providing a demonstration of the existence of a sequence of zeros ${z_k}$, where $lim operatorname{Re}(z_k) = 1$. The exploration of these mathematical phenomena contributes to our understanding of complex analysis and the behavior of the zeta function on the critical line.
Category: Number Theory
[20] viXra:2311.0118 [pdf] replaced on 2025-04-21 00:37:24
Authors: Bertrand Wong
Comments: 14 Pages.
This paper discusses the distribution of the non-trivial zeros of the Riemann zeta function ζ. It looks into the question of whether any non-trivial zeros would ever possibly be found off the critical line Re(s) = 1/2 on the critical strip between Re(s) = 0 and Re(s) = 1, e.g., at Re(s) = 1/4, 1/3, 3/4, 4/5, etc., and why all the non-trivial zeros are always found at the critical line Re(s) = 1/2 on the critical strip between Re(s) = 0 and Re(s) = 1 and not anywhere else on this critical strip, with the first 1013 non-trivial zeros having been found only at the critical line Re(s) = 1/2. It should be noted that a conjecture, or, hypothesis could possibly be proved by comparing it with a theorem that has been proven, which is one of the several deductions utilized in this paper. Through these several deductions presented, the paper shows how the Riemann hypothesis may be approached to arrive at a solution. In the paper, instead of merely using estimates of integrals and sums (which are imprecise and may therefore be of little or no reliability) in the support of arguments, where feasible actual computations and precise numerical facts are used to support arguments, for precision, for more sharpness in the arguments, and for "checkability" or ascertaining of the conclusions. [Published in international mathematics journal.]
Category: Number Theory
[19] viXra:2311.0105 [pdf] replaced on 2024-09-04 20:22:22
Authors: Wiroj Homsup, Nathawut Homsup
Comments: 9 Pages.
The Collatz conjecture considers recursively sequences of positive integers where n is succeeded by n/2 , if n is even, or (3n+1)/2 , if n is odd. The conjecture states that for all starting values n the sequence eventually reaches the trivial cycle 1, 2, 1, 2u2026u2026The inverted Collatz sequences can be represented as a tree with 1 as its root node. In order to prove the Collatz conjecture, one must demonstrate that the tree covers all positive integers. In this paper, we construct a Collatz tree with 1 as its root node by connecting infinite number of basic trees. Each basic tree relates to each positive integers. We prove that a Collatz tree is a connected tree and covers all positive integers.
Category: Number Theory
[18] viXra:2311.0094 [pdf] submitted on 2023-11-20 20:05:24
Authors: Michael Graham
Comments: 3 Pages.
The current Multiplication and Division Properties of Zero are flawed and illogical. This paper illustrates why and presents logical solutions that resolve the issue of dividing by zero.
Category: Number Theory
[17] viXra:2311.0086 [pdf] submitted on 2023-11-19 02:46:21
Authors: Herbert Batte, Florian Luca
Comments: 14 Pages.
Let $(L_n^{(k)})_{ngeq 2-k}$ be the sequence of $k$--generalized Lucas numbers for some fixed integer $kge 2$ whose first $k$ terms are $0,ldots,0,2,1$ and each term afterwards is the sum of the preceding $k$ terms. For an integer $m$, let $P(m)$ denote the largest prime factor of $m$, with $P(0)=P(pm 1)=1$. We show that if $n ge k + 1$, then $P (L_n^{(k)} ) > (1/86) log log n$. Furthermore, we determine all the $k$--generalized Lucas numbers $L_n^{(k)}$ whose largest prime factor is at most $ 7$.
Category: Number Theory
[16] viXra:2311.0070 [pdf] submitted on 2023-11-12 21:44:27
Authors: A. Makarenko
Comments: 4 Pages.
The Collatz algorithm is rewritten to remove divisions by two and to transform it from a hailstone to a steadily growing value. In contrast with the original problem this new sequence becomes reversible and it is reverted in combinatorial way to find all integers leading to the sequence end. Computer programs are available for demonstrations and experimenting.
Category: Number Theory
[15] viXra:2311.0063 [pdf] submitted on 2023-11-10 23:22:13
Authors: Tae Beom Lee
Comments: 5 Pages.
Fermat's Last Theorem(FLT) states that there is no natural number set {a,b,c,n} which satisfies a^n+b^n=c^n or a^n=c^n-b^n when n≥3. In this thesis, we related LHS and RHS of a^n=c^n-b^n to the constant terms of two monic polymials x^n-a^n and x^n-(c^n-b^n). By doing so, we could inspect whether these two polynomials can be identical when n≥3, i.e., x^n-a^n=x^n-(c^n-b^n), which satisfies a^n=c^n-b^n. By inspecting the properties of two polynomials such as factoring, root structures and graphs, we found that x^n-a^n and x^n-(c^n-b^n) can’t be identical when n≥3, except when trivial cases.
Category: Number Theory
[14] viXra:2311.0059 [pdf] replaced on 2023-11-24 16:16:41
Authors: Julian Beauchamp
Comments: 4 Pages.
There are known to exist a number of (multiplicative) cyclic numbers, but in this paper I introduce what appears to be a new kind of number, which we call divisible cyclic numbers (DCNs), examine some of their properties and give a proof of their cyclic property. It seems remarkable that I can find no reference to them anywhere. Given their simplicity, it would be extraordinary if they were hitherto unknown.
Category: Number Theory
[13] viXra:2311.0056 [pdf] submitted on 2023-11-10 01:13:32
Authors: Juan Elias Millas Vera
Comments: 2 Pages.
This paper is about hyperoperators. In this paper I ask myself and the mathematical community if there is possible that a k-ation of the number 2 will be always a number prime for any number m if we add the number one to the result.
Category: Number Theory
[12] viXra:2311.0052 [pdf] replaced on 2024-01-20 01:00:27
Authors: John Yuk Ching Ting
Comments: 82 Pages. Now incorporating Hodge conjecture, Grothendieck period conjecture and Pi-Circle conjecture
We validly ignore even prime number 2. Based on all arbitrarily large number of even prime gaps 2, 4, 6, 8, 10...; the complete set and its derived subsets of Odd Primes fully comply with Prime number theorem for Arithmetic Progressions. With this condition being satisfied by all Odd Primes, we argue that Modified Polignac's and Twin prime conjectures are proven to be true when these conjectures are treated as Incompletely Predictable Problems. In so doing [and with Riemann hypothesis being a special case], this action also support the generalized Riemann hypothesis formulated for Dirichlet L-function. By broadly applying Hodge conjecture, Grothendieck period conjecture and Pi-Circle conjecture to Dirichlet eta function (which acts as proxy function for Riemann zeta function), Riemann hypothesis is separately proven to be true when this hypothesis is treated as Incompletely Predictable Problem.
Category: Number Theory
[11] viXra:2311.0050 [pdf] submitted on 2023-11-08 21:38:34
Authors: John Yuk Ching Ting
Comments: 69 Pages.
As two different but related infinite-length equations through analytic continuation, Hasse principle is satisfied by Riemann zeta function as a certain type of equation that generates all infinitely-many trivial zeros but this principle is not satisfied by its proxy Dirichlet eta function as a dissimilar type of equation that generates all infinitely-many nontrivial zeros. Based on two seemingly different location that are in fact identical, all nontrivial zeros are mathematically located on critical line or geometrically located on Origin point. Thus we prove location for complete Set nontrivial zeros to be critical line confirming Riemann hypothesis to be true. Sieve of Eratosthenes as a certain type of infinite-length algorithm is exactly constituted by an Arbitrarily Large Number of (self-)similar infinite-length sub-algorithms that are specified by every even Prime gaps. Modified Hasse principle is satisfied by this algorithm and its sub-algorithms that perpetually generate the Arbitrarily Large Number of all Odd Primes. Thus we prove Set even Prime gaps with corresponding Subsets Odd Primes all have cardinality Arbitrarily Large in Number confirming Modified Polignac's and Twin prime conjectures to be true.
Category: Number Theory
[10] viXra:2311.0049 [pdf] replaced on 2024-03-27 05:58:56
Authors: A. A. Frempong
Comments: 12 Pages. Copyright © by A. A. Frempong
Collatz conjecture states that beginning with a positive integer, if one repeatedly performs the following operations to form a sequence of integers, the sequence will eventually reach the integer one; the operations being that if the integer is even, divide it by 2, but if the integer is odd, multiply it by 3 and add one; and also, use the result of each step as the input for the next step One would note the patterns of the sequence terms as the Collatz process reaches the equivalent powers, 2^(2k) (k = 2, 3,...) and the sequence reaches the integer 1 by repeated division by 2. Two main cases are covered. In Case 1, the integer can be written as a power of 2 as 2^(k) (k=1,2,3,u2026), and in this case, the sequence would reach the integer one by repeated division by 2, i.e., 2^(k-1), 2^(k-2), 2^(k-3,),u2026,2^(k-k). In Case 2, the integer cannot be written as a power of 2, but the sequence terms reach the equivalent power, 2^(2k) (k = 2, 3,...) and by repeated division by 2, the sequence will reach the integer 1. In Case 2, when the sequence terms reach some particular integers such as 5, 21 and 85, the application of 3n + 1 to these integers will result in the powers, 2^(2k). One would call these integers, the 2k-power converters. There are infinitely many 2k-power converters as there are 2^(2k) powers. There are infinitely many paths for converting integers to 2^(2k) powers. Of these paths, the integer 5-path, is the nearest 2^(2k) converter path to the integer 1 on the 2^(2k)-route. Other integers can follow the integer 5-path to 16 as follows: Let n be an integer whose sequence terms would reach 16, and let n ± r = 5, where r is the net change in the sequence terms before the integer 5; and one uses the positive sign if n<5, but the negative sign if n > 5. One will call the following, the 5-path 2k-converter formula: 3(n ± r) + 1 = 16. By the substitution axiom, using this formula, the sequence of every positive integer that cannot be written as a power of 2, would reach the integer, 16, and continue to reach the integer 1. Therefore, the sequence of every positive integer would reach the integer 1.
Category: Number Theory
[9] viXra:2311.0047 [pdf] submitted on 2023-11-08 21:32:28
Authors: Ángeles Jimeno Yubero, Óscar E. Chamizo Sánchez
Comments: 5 Pages.
Mirror composite numbers are composite numbers of the form 2n-p for some n positive natural number and p prime. We shall show that the factorization of these numbers have interesting properties in order to face the Goldbach conjecture by the divide et impera method.
Category: Number Theory
[8] viXra:2311.0040 [pdf] submitted on 2023-11-08 04:08:29
Authors: Timothy W. Jones
Comments: 3 Pages.
Using a simple application of the mean value theorem, we show that rational powers of e are irrational.
Category: Number Theory
[7] viXra:2311.0030 [pdf] submitted on 2023-11-06 14:27:29
Authors: Timothy W. Jones
Comments: 6 Pages.
Using techniques that show show that e and pi are transcendental, we give a short, elementary proof that pi is irrational based on Euler's formula. The proof involves evaluation of a polynomial using repeated applications of Leibniz formula as organized in a Leibniz table.
Category: Number Theory
[6] viXra:2311.0026 [pdf] submitted on 2023-11-07 01:27:25
Authors: HaeRyong Kim, HyonChol Kim, YongHun Jo
Comments: 13 Pages.
In this paper, we present a new method of evaluating the convergence and sum of a series with the Riemann zeta function in its general term.We consider the convergence and sum of a series by means of difference other than previous methods.
Category: Number Theory
[5] viXra:2311.0025 [pdf] submitted on 2023-11-07 02:14:45
Authors: Ji-Song Ro, Myong-Hyok Sin, Yong-Ho Kim, Sung-Il Gang
Comments: 12 Pages.
We model the rotation process of the motor for variable-order fractional control, which has been active in recent research, and perform numerical simulation of its optimal control and automatic control process. In this paper, we verify numerical method and error estimation of variable order fractional linear dynamic system with time-varying coefficients, a variable-order fractional PID controller design method where the integral of the absolute error with time weight is minimized is proposed using particle swarm optimization algorithm and demonstrate its effectiveness through numerical simulation for DC motor speed control. Numerical experiments show that the performance of the VFPID controller is superior to PID and FPID, especially VFPIDB (B-type variable order FPID) controller has the best performance. Finally, when the differential order varies, the subtypes of variable-order fractional derivatives are analyzed for the effects on the control objective, its effectiveness is newly clarified, and their research and practice is highlighted.
Category: Number Theory
[4] viXra:2311.0015 [pdf] submitted on 2023-11-03 09:25:52
Authors: Waldemar Puszkarz
Comments: 12 pages. Originally posted on ResearchGate in September 2023.
We study the distribution of superprimes, a subsequence of prime numbers with prime indices, mod 4. Rather unexpectedly, this subsequenceexhibits a reverse Chebyshev bias: terms of the form 4k + 1 are more common than those of the form 4k + 3, whereas the opposite is the case in the sequence of all primes. The effect, while initially weak and easy to overlook, tends tobe several times larger than the Chebyshev bias for all primes for samples of comparable size, at least, by one simple measure. By two other measures, it can be seen as fairly strong; by the same measures the ordinary Chebyshev effectis very strong. Both of these measures also imply that the reverse Chebyshev bias for superprimes is more volatile than the ordinary Chebyshev bias.
Category: Number Theory
[3] viXra:2311.0010 [pdf] submitted on 2023-11-04 00:20:41
Authors: Carlos Villacres
Comments: 9 Pages.
An approach to the classic problem of Fermat's last theorem. Using the binomial theorem and the cases where n is even or odd, we find a solution as well as a Pythagorean triple generator.
Category: Number Theory
[2] viXra:2311.0006 [pdf] submitted on 2023-11-03 03:13:33
Authors: Yajun Liu
Comments: 11 Pages.
In this paper, we discuss the symmetry of N-domain and we find that using the symmetry characters of Natural Numbers we can give proofs of the Prime Conjectures: Goldbach Conjecture、Polignac’s conjecture (Twins Prime Conjecture) and Riemann Hypothesis. . We also gave a concise proofs of Collatz Conjecture in this paper.
Category: Number Theory
[1] viXra:2311.0003 [pdf] submitted on 2023-11-01 21:26:30
Authors: Jim Rock
Comments: 2 Pages.
In 1742 Christian Goldbach suggested that any even number four or greater is the sum of two primes. The Goldbach Conjecture remains unproven to the present day though it has been verified for all even numbers up to 4 x 1018. This paper suggests an algorithm for checking the Goldbach conjecture for individual even numbers and a generalization that could be used to prove the Goldbach conjecture.
Category: Number Theory
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