| viXra.org > Number Theory > 2502 |
 |
 |
| viXra.org > Number Theory > 2502 |
|
|
[28] viXra:2502.0196 [pdf] submitted on 2025-02-28 20:58:22
Authors: Oliver Couto
Comments: 3 Pages.
There are numerical solutions available on Wolfram world of mathematics website (ref. # 4) for the equation (p^6+q^6+r^6+s^6)=2(a^6+b^6). In this paper the author has arrived at numerical solution by algebra. It is common knowledge that arriving at numerical solutions by algebra is difficult for degree four & above. Also on the internet the author has not come across any method for the above mentioned equation.
Category: Number Theory
[27] viXra:2502.0195 [pdf] submitted on 2025-02-28 22:23:30
Authors: Bahbouhi Bouchaib
Comments: 21 Pages. Interested readers can cite this article which is now published in J Math Techniques Comput Math 4(1) 2025 (opast publishing group)
The main idea of this article lies in the fact that Goldbach's strong conjecture is associated with the progression ofnatural integers from 0 to infinity, which results in precise gaps between prime numbers. The gap of 6 is the mostregular between primes 6x + 1 on the one hand and primes 6x — 1 on the other. In this article, using the equations 3x ± 5and analyzing the 6-based gaps between primes while determining the initial conditions that make a prime appear afteror before an integer, this article argues for the truth of Goldbach's strong conjecture. Two new concepts are introducedfor the first time : Goldbach's gap and Goldbach's transposition. By analyzing its key digits (units and tens), a primenumber itself can lead to the conversion of an even number into two primes. A new algorithm is deduced from theseresults, enabling us to locate prime numbers located at equal distance from any integer, even or odd, prime orcomposite. This constitutes a decisive proof of Goldbach's strong conjecture, since it means that any even number canbe converted into the sum of two prime numbers.
Category: Number Theory
[26] viXra:2502.0186 [pdf] submitted on 2025-02-27 15:42:14
Authors: James DeCoste
Comments: 20 Pages. Contact: jbdecoste@eastlink.ca
After dissecting the mechanics of locating valid twin primes, I was able to establish a Proof through contradiction. I start by creating a table to easily display the potential list of twin primes. Using an elimination matrix scheme, I systematically remove twin prime candidates from the list if either half of the pair are multiples of an already known 'Prime Number'. Multiples of prime numbers, primes squared and primes multiplied by other primes are not prime numbers themselves (examples 5*5=25; 5*7=35; 5*11=55; 7*7=49; 7*11=77; and so on). It's an easy approach with repeatable patterns for each prime number. It quickly becomes obvious that these elimination patterns are repeating for all non-prime removals. All these elimination patterns are of the form remove-skip(n)-remove-skip(m)...repeated to infinity. Note that n+m+2 is the prime number. The first non-prime removal for any prime is in essense that prime^2 (prime squared). A prime number squared will always fall into the sixth column ( the column starting with 7)! Further, two adjacent patterns will slightly overlap if those two primes form a twin prime pair. I then proceed to make the 'silly' assumption that there will be no potential twin prime candidates in the initial skip(n) plus skip(m) regions for the two overlapped twin primes (entire initial pattern for a given twin prime pair) in this elimination matrix. If we assume that 11 & 13 are the last twin primes possible, we would have to make the assumption that there are no twin primes candidates in the elimination overlapped pattern regions for either prime 11 or 13 combined at minimum. The contradiction arises because we can show that there is always at least one twin prime pair in this combined/overlapped region. As long as there are infinitely many primes there will be infinitely many twin primes. Euclid proved there are infinitely many primes with his proof. I have simply extended his proof into my own.
Category: Number Theory
[25] viXra:2502.0178 [pdf] replaced on 2025-04-01 22:19:43
Authors: Oreste Caroppo
Comments: 15 Pages.
In this article we will propose initially a graphical method for carrying out the algorithm which is at the base of the famous Collatz Conjecture; therefore we will demonstrate analytically that, for no natural number, the Collatz Algorithm can present an infinite sequence of ever-increasing steps. The development of the graphical method will lead us to extrapolate a simplified and more basic scheme of the Collatz Conjecture, which will allow us in turn to consider the Collatz Conjecture Algorithm as a sorte of particular case of a more general algorithm valid for positive real numbers, which, for statistical reasons that we will show, will always converge towards the smallest real number, zero. The Collatz Algorithm will therefore appear as an adaptation only to the natural numbers of this more general algorithm, which is why Collatz Algorithm also tends to converge towards the smallest natural number which is 1.
Category: Number Theory
[24] viXra:2502.0173 [pdf] replaced on 2026-03-14 21:05:43
Authors: Johan Noldus
Comments: 143 Pages.
We covariantize quantum field theory and extend quantum theory to the spiritual domain required for structure formation.
Category: Number Theory
[23] viXra:2502.0172 [pdf] replaced on 2025-08-08 07:17:05
Authors: Emmanuil Manousos
Comments: 29 Pages.
In this paper we prove an original theorem about the natural numbers. From this theorem we obtain a set of symmetries related to the factors of the natural numbers. We study these symmetries. We derive a set of conclusions about the structure of the natural numbers. Among them is an algorithm for factoring odd numbers.
Category: Number Theory
[22] viXra:2502.0164 [pdf] submitted on 2025-02-24 03:21:36
Authors: Suryansh Singh Shekhawat
Comments: 5 Pages.
This paper presents a computational investigation that searches for arithmetic expressions approximating a given target constant using a finite set of base expressions and allowed operators. In our approach, the search is organized by a notion of DEPTH (i.e., the number of operations applied) and is limited to a maximum depth (typically 10 or 11) due to computational constraints. We describe the algorithm rigorously, introduce precise definitions and notation, present pseudocode in the algorithmic style, and discuss sample solutions—including approximations for e, φ (the golden ratio), and π with their respectivedepths and computed absolute errors. We also include an example run showing the number ofcandidate expressions generated at each depth (with depth 10 evaluating approximately 2.5 million sequences). Finally, we discuss the inherent limitations, including memory (RAM) requirements for deeper searches, and how increased computational power could extend the search depth.
Category: Number Theory
[21] viXra:2502.0161 [pdf] submitted on 2025-02-24 02:53:42
Authors: Jau Tang
Comments: 8 Pages.
We present a simple rigorous proof of Riemann’s hypothesis. This hypothesis has remained unsolved since Riemann’s original formulation in 1859, although numerous zeros have been found along the critical line with the assistance of computer calculations. Our analytic proof is based on the analysis of the reflection symmetry between |Γ(su20442))〖(ζ(s))u2044π^(su20442) |〗^2 and |Γ(((1-s))u20442)) 〖ζ(1-s)u2044π^(((1-s))u20442) |〗^2, although the zeta and Gamma functions are asymmetric. We show their global minimum along the x-direction throughout the critical strip, their zeros, and the non-trivial zeros of the zeta function must occur at s=1/2+iy. If the zeros were not along the critical line, we show contradictions to the properties of the symmetric functional pair would arise. Thus, we prove rigorously the validity of Riemann’s conjecture.
Category: Number Theory
[20] viXra:2502.0155 [pdf] submitted on 2025-02-23 01:54:59
Authors: Samuel Bonaya Buya
Comments: 8 Pages.
This paper presents an insightful relationship between arithmetic and Geometric mean. A bridge is established between arithmetic, geometric and harmonic mean. The concept is useful in number theory since Goldbach conjecture also implies that all integers greater than 1 are an arithmeticmean of a pair of primes.
Category: Number Theory
[19] viXra:2502.0124 [pdf] replaced on 2025-02-22 05:23:55
Authors: Minhyeok An
Comments: 16 Pages. If this paper contains no errors, resolving Bunyakovsky’s conjecture through this approac.h would hold even greater mathematical significance than previously recognized.
The numbers whose sum of divisors is a perfect square may initially appear to follow no specific pattern. However, through this research, I have identified a particular rule related to prime numbers. Furthermore, I establish that the existence of infinitely many numbers whose sum of divisors is a perfect square is a necessary and sufficient condition for the existence of an irreducible polynomial with integer coefficients that generates infinitely many prime numbers. Additionally, I explore its connection to Bunyakovsky's conjecture.
Category: Number Theory
[18] viXra:2502.0120 [pdf] submitted on 2025-02-17 20:47:41
Authors: Ruslan Pozinkevych
Comments: 5 Pages.
One of the biggest issues in discrete mathematical research is to set up the mapping system that would clearly define the transition between various counting groups e.g decimal, ternary, nonary etc. This is only possible if we set a one-to-one correspondence between members of a set In his previous work "Optimization Techniques in Ternary Calculus" the author just tried to do that by relating decimal numbers to their binary correspondence however a universal formula is much more desirable and current investigation will attempt to find this very universal approach$left[ 8 ight]$
Category: Number Theory
[17] viXra:2502.0112 [pdf] submitted on 2025-02-16 13:13:19
Authors: Reece Tinkler
Comments: 5 Pages.
Pythagorean triples are sets of three whole numbers that satisfy the equation. This paper introduces two new formulas related to Pythagorean triples. The first formula determines the largest possible value for a given, ensuring that all elements remain whole numbers. The second formula explores the special case where, identifying recurrent patterns within Pythagorean triples.
Category: Number Theory
[16] viXra:2502.0102 [pdf] submitted on 2025-02-14 11:24:45
Authors: Fabrice Trifaro
Comments: 16 Pages.
The Collatz conjecture, also known as the Syracuse conjecture or the 3x+1 problem, is a mathematical conjecture according to which the Collatz sequence always reaches the value 1, and then repeats the cycle (1,4,2) indefinitely, regardless of the first term of the sequence as long as it is a strictly positive integer. It originated in the 1930s and its authors are mainly Lothar Collatz and Helmut Hasse, the latter shared it in the United States during a visit to Syracuse University, and the Collatz sequence then became known as the Syracuse sequence. To date, this conjecture has not been proven either true or false. The purpose of this study is to prove, as clearly and precisely as possible, that this conjecture is true. The proof is based on classical mathematics which should not pose any major difficulties.
Category: Number Theory
[15] viXra:2502.0089 [pdf] submitted on 2025-02-13 21:26:40
Authors: Ahcene Ait Saadi
Comments: 03 Pages.
In this paper I highlight the wonders of natural integers. I discovered pairs of integers which, by doing operations on them, I find relationships between the parts of these numbers. I put this work in the hands of young researchers for deepening, this in the interest of science and knowledge.
Category: Number Theory
[14] viXra:2502.0083 [pdf] submitted on 2025-02-12 21:03:00
Authors: Ahcene Ait Saadi
Comments: 09 Pages.
In this paper I solve Diophantine equations using the pseudo differential calculation of Ait Saadi. I solve Diophantine equations of the form: ax+by=c without using Euclid's algorithm. Finally I give the general method of solving the equations of the form ax+by+cz=d.
Category: Number Theory
[13] viXra:2502.0066 [pdf] submitted on 2025-02-11 23:11:27
Authors: Suryansh Singh Shekhawat
Comments: 5 Pages.
In the decimal number system we can find some interesting products, such as 12 = 3 · 4. What is interesting about these products is that if you remove the symbols, the digits form a sequence 1, 2, 3, 4. Another example is 56 = 7 · 8, where the sequence is 5, 6, 7, 8. The objective of this paper is to prove that, within the constraints of the decimal number system, these are the only two cases in which this happens. In this paper we also consider a general case which considers subsequences of mod 10, i.e., 0123456789012345..
Category: Number Theory
[12] viXra:2502.0065 [pdf] submitted on 2025-02-11 23:09:41
Authors: Suryansh S. Shekhawat
Comments: 14 Pages.
his paper considers prime numbers as a sequence that can be described by arithmetic progressions missing on a few terms. This tool (cancelling sequences) can be used to generalise the sieves of Eratosthenes, Sundaram, etc, and resolve them into generating formulae with a few unknowns.
Category: Number Theory
[11] viXra:2502.0064 [pdf] replaced on 2025-02-15 06:18:48
Authors: Samuel Bonaya Buya
Comments: 14 Pages. Please replace previous version with this
In this research a formulation of the approximate sum of primes is presented.With it also is presented the mean of primes. The formulations are used to confirmthe validity of the Binary Goldbach conjecture through establishing the intervalcontaining Goldbach partition primes of a set of even numbers. A general Goldbachpartition theorem is established by which Goldbach conjecture is proved. Finallythe paper seeks to get a better prime counting function than li (x)
Category: Number Theory
[10] viXra:2502.0056 [pdf] submitted on 2025-02-08 21:53:01
Authors: Laurent Nedelec
Comments: 37 Pages.
After introducing definitions related to the Collatz problem (Part 1), the concept of "verified integers" and several organizational rules around this concept are presented (Part 2). A unique logical tool, the axis of verified integers, is highlighted (Part 3). In Part 4 , it is proven that all bounded trajectories without non-trivial cycles are verified. These elements allow the development of a systematic approach to solving the Collatz problem with the help of inverse graphs (Part 5). The issue of non-trivial cycles and divergent trajectories is then explored (Parts 6 and 7). Ultimately, we arrive at two contradictory propositions :1) Either all integers satisfy the Collatz conjecture, or2) An infinite number of integers do not satisfy it. This eliminates the possibility that only a small number of integers fail to satisfy the conjecture, while the rest do. The conclusion of this study leans toward the first solution : all integers satisfy the Collatz conjecture.This text, written in English, is a translation of the original French text published five months ago on Vixra (Vixra 2410.0063) under the title "New tools to verify the Collatz conjecture". During the translation process, Parts 5, 6 and 7 were extensively revised, resulting in a new text that differs from the initial French version.
Category: Number Theory
[9] viXra:2502.0054 [pdf] replaced on 2025-08-27 20:43:48
Authors: Liao Teng
Comments: 88 Pages.
In order to strictly prove the conjecture in Riemann's 1859 paper on the Number of prime Numbers Not Greater than x from a purely mathematical point of view, and strictly prove the correctness of Riemann's conjecture, this paper uses Euler's formula to prove that if the independent variables of ζ(s) function are conjugate, then the values of ζ(s) function are also conjugate, thus obtaining that the independent variables of ζ(s) function are also conjugate at zero. And using the conjugation of the zeros of the Riemann ζ(s) function and the zeros of ζ(s)=0 and the zeros of ζ(1-s)=0, s and 1-s must also be conjugated, The nontrivial zero of Riemann function ζ(s) must meet s=1/2 +ti(t∈R and t≠0) and s=1/2-ti(t∈R and t≠0). And the symmetry of the zeros of Riemann ζ(s) function is the necessary condition that the nontrivial zeros of Riemann ζ(s) function are located on the critical boundary. According to the symmetry property of the zeros of Riemann ζ (s) function s and the zeros of Riemann ζ(s) function 1-s, combined with the conjugated property of the zeros of Riemann ζ(s) function s and Riemann ζ(s) function 1-s, It is shown that the real part of the nontrivial zero of the ζ(s) function must only be equal to 1/2. And by Riemann set s=1/2+ti(t∈C and t≠0) and auxiliary function ξ(s)=1/2.s(s-1)Γ(s/2)π^(- s/2) ζ(s)(s∈C and s≠1), Get∏s/2(s-1)π^(-s/2)ζ(s)=ξ(t)=0, combining the nontrivial zeros of Riemann function ζ(s) must meet s=1/2+ti(t∈R and t≠0) and s=1/2-ti(t∈R and t≠0), Thus it is proved equivalently that the zeros of the Riemann ξ(t) function must all be non-zero real numbers, and the Riemannian conjecture is completely correct.
Category: Number Theory
[8] viXra:2502.0052 [pdf] submitted on 2025-02-08 21:26:54
Authors: Silvano Mattioli
Comments: 2 Pages. (Note by viXra Admin: Please cite and list scientific references)
In this paper we want to demonstrate the inconsistency of the ZFC axioms as expressed. Starting from the definition of "non-computable number" and using the mapping of the Peano curve we arrive at the absurdity of generating a non-computable number starting from an algorithm.This conclusion is evidently in contrast with the definition of a non-computable number and therefore creates the paradox that causes the castle of ZFC axioms to collapse.
Category: Number Theory
[7] viXra:2502.0033 [pdf] replaced on 2025-02-10 16:51:41
Authors: SeongJoo Han
Comments: 16 Pages.
We found why "Eternal Loop" can't exist in Multi-Dimensional space, and why any number can't diverge. And "Collatz Tree" includes most things in the world as indexed. It seems like a sleeping lion.
Category: Number Theory
[6] viXra:2502.0027 [pdf] submitted on 2025-02-05 21:55:36
Authors: Ahcene Ait Saadi
Comments: 9 Pages.
1) In this article, I study prime numbers from polynomials that generate prime numbers. I have put the list of these polynomials and I have applied this method to some polynomials. 2) I used an empirical method to find series of sum of numbers that give a sum of prime numbers. this is just the beginning, I hope that young researchers will deepen this work for the benefit of science and knowledge.
Category: Number Theory
[5] viXra:2502.0009 [pdf] submitted on 2025-02-01 05:47:33
Authors: Samuel Bonaya Buya
Comments: 8 Pages. This paper was written in 2018 and its ideas need to be preserved
A simple proof of the Legendre conjecture is presented by introducing a suspected prime counting function
Category: Number Theory
[4] viXra:2502.0007 [pdf] submitted on 2025-02-01 19:59:29
Authors: Samuel Bonaya Buya
Comments: 3 Pages.
A proof by implication method is introduced for proving the Legendre conjecture.
Category: Number Theory
[3] viXra:2502.0005 [pdf] submitted on 2025-02-01 20:00:29
Authors: Samuel Bonaya Buya
Comments: 3 Pages.
This research aims at coming up with some formulation of the interval containing a given number of primes using the prime number theorem.
Category: Number Theory
[2] viXra:2502.0002 [pdf] replaced on 2025-02-06 22:21:32
Authors: Bahbouhi Bouchaib
Comments: 18 Pages. This is a new attempt to demonstrate Goldbach's strong conjecture using basic and accessible mathematics.
After defining the writing of even numbers with equations 4x ± 1; the article shows that odd numbers obey loops of 4 numbers to infinity that obey the equations 4x ± 1. Each odd number in the loop can be prime (P) or composite (C), but for an even number E to be E = P + P' such that P' > P and such that P' > E/2 and P < E/2, P and P' must belong to two loops symmetrical with respect to E/2 and occupy the same positions in them and specific unit digits. The article shows that the three possible sums of an even number E are E = C + C'; or E = P + C; or E = P + P' (C is composite and P is prime). The article demonstrates that E = C + C' ↔ E = C + P ↔ E = P + P' ; and that one sum can be converted into another by subtracting and adding gaps of 4n from or to the two terms of addition. This is a deductive demonstration of Goldbach's strong conjecture shown here for the first time.
Category: Number Theory
[1] viXra:2502.0001 [pdf] replaced on 2025-03-12 22:11:41
Authors: Jabari Zakiya
Comments: Significantly updated paper (4 extra pages), with new findings, optimized software code, and more and extensive data.
Goldbach’s Conjecture states every even integer n > 2 can be written as the sum of 2 primes, while Bertrand’s Postulate states for each n ≥ 2 there is at least one prime p such that n < p < 2n. I show both are essentially statements on the distribution of primes, and their inherent properties when modeled and understood as the residues of modular groups Zn. In addition, a much tighter dynamic bound on p than given by the BP will be presented.
Category: Number Theory
| Third-Party Links: |
|