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[18] viXra:2301.0154 [pdf] replaced on 2023-08-30 08:53:46
Authors: Xingyuan Zhang
Comments: 21 Pages. This article did proved Collatz conjecture. Method is reverse tracing. My conclusions and middle results are very beautiful. If interested, to discuss, or even cooperate to finish the paper for submission!
In this paper we had given an elementary proof of the Collatz conjecture, it holds. By detailed analysis of the properties of both forward and inverse operations of the proposition, we had some important conclusions: 1, there are no cycles except 1 to 1, and for a given odd it either goes to infinity or returns to 1 in forward operations; 2, there hasn’t any triple in the forward path numbers; 3, there have an infinity number of inverse path numbers which had been defined as similar numbers between them in one time of inverse operation; 4, to do inverse operations (defined as reverse tracing) repeatedly from the odd 1, it will obtain all of the odds; 5, for any odd obtained by tracing, to do forward operations, it must return to 1 along the reverse tracing path.
Category: Number Theory
[17] viXra:2301.0152 [pdf] submitted on 2023-01-30 01:47:33
Authors: Roberto Violi
Comments: 18 Pages.
In this article, we solve one of the oldest and most celebrated problems in number theory, namely the existence or nonexistence of odd perfect numbers. We know there is no number of this type having less than 100 digits. A number is said to be perfect if it is the sum of its proper divisors. Euclid in his The Elements ninth book gives a formula for all even perfect numbers. We answer the question of whether there exists an odd perfect number in the negative by proving a theorem asserting that the existence of such a number would lead to contradictions (proof by reductio ad absurdum). Somewhat remarkably, perhaps, this result is proved using only elementary methods. Hence, the popular conjecture that odd perfect numbers do not exist, no matter how large these numbers might be, is confirmed to be correct. Thus, one of the oldest and most celebrated questions in mathematics has now a definitive answer.
Category: Number Theory
[16] viXra:2301.0149 [pdf] submitted on 2023-01-28 19:10:35
Authors: Edgar Valdebenito
Comments: 3 Pages.
We give some formulas related to (2^(3/4))*F(-1/8,1/4,5/4,1), where F is the Gauss hypergeometric function.
Category: Number Theory
[15] viXra:2301.0141 [pdf] replaced on 2023-02-08 10:05:43
Authors: Muhammad Ameen Ben Hmida
Comments: 4 Pages. Copyright belongs to the author
This is an algebra-based proof of Goldbach's conjecture. First, we define the formula for prime numbers greater than 3, then we prove that every even number greater than 10 can be written in the form of the sum of two prime numbers, and finally we deal with the rest of the even numbers less than 10, which are 8 , 6 and 4 .
Category: Number Theory
[14] viXra:2301.0136 [pdf] replaced on 2023-01-27 13:39:25
Authors: Zhi Li, Hua Li
Comments: 13 Pages.
To calculate the number of prime numbers, the prime number theorem function x/ln (x) and Gaussian function Li (x) are most commonly used. However, the former is always less than andthe latter is always greater than the actual number of prime numbers, and the deviation increases with the increase of the order of magnitude. The function p (x) has been proposed to improve x/ln (x). Now the Gaussian function Li (x) is dynamically modified, and a more ideal prime number estimation function q (x) is obtained. Numerical experiments show that the modified q (x) calculation is simple and accurate compared with the calculation results of p (x) and Riemann function R (x).
Category: Number Theory
[13] viXra:2301.0133 [pdf] submitted on 2023-01-25 18:02:43
Authors: Edgar Valdebenito
Comments: 2 Pages.
We give some formulas related to gamma=0.577215..., here gamma is the Euler's constant.
Category: Number Theory
[12] viXra:2301.0123 [pdf] submitted on 2023-01-22 06:33:42
Authors: Juan Elias Millas Vera
Comments: 7 Pages.
In this paper I give a counterexample of the Eisenstein conjecture which said that the tetration of number 2 plus 1 gives always a prime number.
Category: Number Theory
[11] viXra:2301.0121 [pdf] submitted on 2023-01-22 14:51:31
Authors: Lee Yiu Sing
Comments: 1 Page.
I will show how to prove and e are irrational numbers with the fact that they are transcendental numbers.
Category: Number Theory
[10] viXra:2301.0119 [pdf] submitted on 2023-01-23 01:35:00
Authors: Enrico D'Ambrosio
Comments: 8 Pages.
In 1637 the Toulouse lawyer Pierre de Fermat enunciated a conjecture in number theory which denied the possibility of dividing the nth power of an integer into the sum of two nth integer powers when natural exponents greater than or equal to three are considered. This is the famous "Fermat's last conjecture", now elevated to a theorem thanks to the proof given by Prof. A. Wiles in 1995. In other words, the Diophantine equation x^n + y^n = z^n with n ≥ 3 does not admit integer, primitive and non-trivial solutions. In my article, I set out my observations on a particular case of this theorem. The "admirable proof" which Fermat claimed to have was never found, except for the one in which n = 4 and which the French genius demonstrated with a new method coined by himself and known as the "method of infinite descent". The latter follows the same logic as the already known "ab absurdum reasoning" even if it is articulated with slightly more demanding algebraic procedures.In this article I report my proof of the case n = 4 with the use of an elementary algebraic procedure which, avoiding Fermat's "infinite descent", brings the equation in question back to that of the well-known quadratic equation:x^2 + 2y^2 = z^2 for which all the primitive, non-trivial solutions in the set of integers have already been parameterized. The peculiarity of my method lies in transforming Fermat's quartic equation into a quadratic equation well studied by number theorists and whose solutions are already known. With this strategy it is possible to prove the thesis.
Category: Number Theory
[9] viXra:2301.0104 [pdf] replaced on 2023-01-27 13:36:08
Authors: Zhi Li, Hua Li
Comments: 12 Pages.
It has been human's dream for thousands of years to find the ideal prime number counting function and prime number universal formula. For the number of prime numbers smaller than the given order of magnitude,the calculation deviation of the prime number theorem function x/ln (x) isrelatively large. However, the calculation with Riemann formula R (x) function is too complex, and it deviates from the true value as GaussianLi (x) function when the given order of magnitude is very large. Now the prime number theorem function x/ln (x) is dynamically modified andfurther optimized by using the calculation formula of Mersenne's prime number, and a simple, easy and relatively accurate calculation formula is found.
Category: Number Theory
[8] viXra:2301.0103 [pdf] submitted on 2023-01-20 20:27:44
Authors: Edgar Valdebenito
Comments: 5 Pages.
In this note we give some integrals for K(sqrt(3)/2).
Category: Number Theory
[7] viXra:2301.0085 [pdf] submitted on 2023-01-18 02:30:18
Authors: Filippo Giordano
Comments: 25 Pages.
The infinite set of natural numbers is formed by infinite subsets of pairs of quadratic intervals [n(n-1)+1, n^2], [n^2+1, n(n+1)]. Each of these pairs of intervals is formed by increasing quantities of elements, all governed by their divisor, of value ≤ n, closest to n. By bringing together these individual divisors of each element, a scale is formed for each interval, including all values included in the interval [1, n]. By assigning the name Mm (Major than minor) to these particular divisors, we note that between the elements and their divisors Mm, there is a one-to-one group correspondence which always allows for each quadratic interval the presence of at least one element having the trivial divisor 1 , i.e. the presence of at least one prime number, as noted by Oppermann's conjecture of 1882. By extending Fermat's method of factoring natural numbers, through quadratic number lines, it is confirmed that each of the divisors Mm finds group correspondence with at least one of the elements of the quadratic intervals. The mathematical law that regulates the distribution of prime numbers, therefore, is expressed through the divisors Mm which, from time to time, depending on the occurrences of each quadratic interval, observe some fundamental rules common to all quadratic intervals. Furthermore, by observing the quadratic intervals arranged inside the Spiral of Ulam, one has the opportunity to observe that, in a fascinating way, Nature places all the quadratic intervals, assembling them according to their four different typologies, each in a different cardinal direction: quadratic intervals A of odd n, quadratic intervals B of odd n, quadratic intervals A of even n, quadratic intervals B of even n.
Category: Number Theory
[6] viXra:2301.0082 [pdf] submitted on 2023-01-17 02:18:13
Authors: Theophilus Agama
Comments: 5 Pages.
We apply the notion of the textbf{olloid} to show that the ErdH{o}s-Straus equation $$frac{4}{n^{2^l}}=frac{1}{x}+frac{1}{y}+frac{1}{z}$$ has solutions for all $lgeq 1$ provided the equation $$frac{4}{n}=frac{1}{x}+frac{1}{y}+frac{1}{z}$$ has solution for a fixed $n>4$.
Category: Number Theory
[5] viXra:2301.0080 [pdf] submitted on 2023-01-17 02:21:45
Authors: Filippo Giordano
Comments: Pages.
The formula with which all the primitive Pythagorean triples are obtained, provided by Euclid around 300 BC, attributes to m, n alternative odd, even values, provided that m and n are coprime, i.e. without common divisors and that m> n. The same formula in cases where m, n, are both odd or both even provides only derived Pythagorean triples. After a specific search I found an alternative form to the algorithm which allows to obtain all the primitive Pythagorean triples a^2+b^2=c^2 by assigning both odd positive integer values to m, n and to obtain primitive Pythagorean triples but with mixed values (integers and decimals ) to a,b,c, assigning to m, n alternative odd, even values.
Category: Number Theory
[4] viXra:2301.0061 [pdf] replaced on 2023-11-13 18:35:32
Authors: James DeCoste
Comments: 59 Pages. Contact: jbdecoste@eastlink.ca
This is a mathematical analysis of everything Collatz. I've come up with a revolutionary way of representing the natural counting numbers as an infinite set of equations. From these I am able to make some provable connections that not only show that all counting numbers are used once in the Collatz Tree structure; but where additional loops originate; the importance of 4x+1 and 2x+1; duality of even numbers; among others. I also show that there can only be one unbroken chain of continuous "3n+1 / 2" growing toward infinite number sizes approaching infinity but never actually getting there. This would be the 'only' counter-example that is possible and as odds would have it, it does not pan out. That only possible counter example is not to be.Using the induction method where we show that x = 1 is true (elementary, since it is part of the initial loop); from there we assume that x from 1 to k are also true building on the x=1 being true; then k+1 is also true. That is a complicated way of saying that if we know and assume all numbers from 1 to k are true, then the very next number k+1 is also true in as much as we apply the two rules correctly so the number reduces to one that is already in the proven set!The first three equations of my infinite set of equations are easy to apply this induction to and cover 87.5% of the counting number set. I change things up a bit for the upper level equations. I am able to prove through the same induction method that any number that is not a multiple of 3 ( falling in these levels/equations ) is also provable. Stepping outside the usual method of this proof I investigate the multiples of three separately to prove they are all following a similar induction proof. And they do. All said and done I am able to prove that 100% are provable. (4x+1) is important in this proof as well the application of (3x+1)/2. Read on to find out what I mean.I've covered off on the loop issue part of the proof by showing how additional loops come about in the Collatz Tree structure. There is only one loop in Collatz ( positive counting numbers ) and that is the trivial { 1 — 4 — 2 } loop. No others are possible no matter how close to infinity one gets and all numbers will reduce to this trivial loop.The detailed discussion of how I arrived at these different conclusions is outlined below. I apologize if some sections are difficult to follow. I am not a mathematician by nature or profession. I do love mathematics though. I hope you enjoy my approach of showing the 'self' enlightening process as I continued to explore. As my expertise improved, other intuitive aspects became readily useful in the proof. The reader will appreciate knowing why I went down each path I chose to pursue.This is an updated version of my original document with a new section near the end that gets into the details of the proof. The remainder of the original report remains intact for the most part but does have additional details and concepts introduced and dispersed therein.This is the third version where I have solved the outstanding subset of multiples of 3. I believe you will find that method eye-opening since it involves some under the sheets number manipulation by multiple applications of (3x+1)/2. I also introduce the 'duality' nature of even numbers that remain hidden in the Collatz tree structure... and that is that those even numbers can behave as if they were odd numbers; (Odd*3)+1=Even; (Even*3)+1=Odd; (4*Even)+1=Odd.After arriving at this proof I go back to my original set of equations to see if they behave the same way and may provide an easier more condensed method to the final proof. Low and behold they do! It boils down to one chart. Now that I see it on paper I am impressed with my progress. It has taken just over 3 years to finalize the process.In this, the forth version, I have looped back to my original infinite set of equations to formulate a single chart from whence a complete proof can be understood. I have also added a 'final' section that simplifies the induction method of all odd numbers. I think you'll be enlightened with that approach since it really simplifies the inductive proof. If I am right this simplified approach can be a much simplified proof in itself.This detailed analysis has led me to two alternate methods for complete proofs. Enjoy.
Category: Number Theory
[3] viXra:2301.0047 [pdf] submitted on 2023-01-07 10:09:35
Authors: Filippo Giordano
Comments: 2 Pages.
Euclid, in 300 BC observed that with n = prime number, whenever 2^n —1 corresponds to a further prime number, then (2^n—1)2^(n—1) is a perfect number.As the research on perfect numbers went on, a curious property of them was noticed: the sum of the single digits of which each perfect number is composed (with the exception of 6), perpetuated until a single digit is reached, always converges to 1. This characteristic leads to the hypothesis that all perfect numbers, including those still unknown, retain this property. But why does the first perfect number not coincide with the root 1? Is it the only exception or will others be discovered later? The glimpse of light that illuminates the conjecture finds an explanation in the cyclical dimensions of numerical systems.
Category: Number Theory
[2] viXra:2301.0032 [pdf] submitted on 2023-01-05 02:50:52
Authors: Leonardo De Lima, Vitor Ermani
Comments: 9 Pages.
This article will prove, by reduction to absurdity, that the zeta function has at least one non-trivial zero outside the critical line.
Category: Number Theory
[1] viXra:2301.0011 [pdf] submitted on 2023-01-02 05:45:51
Authors: Aric B. Canaanie
Comments: 6 Pages. www.0bq.com
The central idea of this article is to introduce and prove a special form of the zeta function as proof of Riemann’s last theorem. The newly proposed zeta function contains two sub-functions, namely f1(b,s) and f2(b,s) . The unique property of zeta(s)=f1(b,s)-f2(b,s) is that as tends toward infinity, the equality zeta(s)=zeta(1-s) is transformed into an exponential expression for the zeros of the zeta function. At the limiting point, we simply deduce that the exponential equality is satisfied if and only if real(s)=1/2. Consequently, we conclude that the zeta function cannot be zero if real(s)=1/2, hence proving Riemann’s last theorem.
Category: Number Theory
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