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[13] viXra:2307.0158 [pdf] submitted on 2023-07-29 21:35:56
Authors: Xiaohui Li
Comments: 2 Pages.
[A]ny large even number can be written as the sum of two prime numbers, which is also called "strong Goldbach's conjecture" or "Goldbach's conjecture about even numbers". Based on the equality of the sum of all odd numbers and the equality of odd numbers, the values and numbers of prime numbers pr1 and pr2 are separately screened out from the odd combination. By using the equality of the sum of all odd numbers and the equality of all odd numbers, an identity is constructed to obtain 2n=pr1+pr2
Category: Number Theory
[12] viXra:2307.0149 [pdf] submitted on 2023-07-27 07:28:28
Authors: Tae Beom Lee
Comments: 5 Pages. (Correction on author name made by viXra Admin - Please conform!)
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 when n≥3. In this thesis, we related the LHS of a^n=c^n-b^n to x^n-a^n and the RHS to x^n-(c^n-b^n). By doing so, we could analyse FLT in view of properties of monic polynomials such as factoring, root structure and graphs.The polynomial properties narrowed the vast possible approaches to FLT to elementary level mathematics. We relied on factoring, rational root theorem and parallel movement of graphs. And we succeeded to find simple proofs of FLT, which many people waited for so long time.
Category: Number Theory
[11] viXra:2307.0137 [pdf] submitted on 2023-07-25 04:11:56
Authors: Hideharu Maki
Comments: 173 Pages.
The functional equation of real variable that Riemann used in his paper was subjected to elementary operations. And I obtained a lot of complex functional equations that the Riemann zeta function follows respectively. Here, functional equation transformations were the main methods for obtaining the complex functional equations. Half of those are equivalent to the complete symmetric functional equation that the Riemann Xi function follows, and one of those has an origin symmetry with correction terms. From the origin symmetric functional equation including correction terms, the representation containing the leading term of the zeta function for any complex number was obtained. And the Riemann hypothesis was proved by applying reduction to absurdity. Moreover the general representation containing the leading term of the zeta function for any odd number of 3 or more was also obtained. By suitably combining those functional equations, I observed a new explicit formula for the zeta function. The Riemann hypothesis was again proven using the deductive method. And two types of general representations for the zeta function for any odd number of either 3 or 7, or more, were also obtained from the explicit formula. In total, three types of general representations for the zeta function for any odd number of either 3 or 7, or more, were discovered. Conversely, I defined a new function, named the Chi function, for the left side of the origin symmetric functional equation that includes corrective terms. The Chi function is similar to the Riemann Xi function and exhibits origin symmetry. Furthermore, I defined a new function, the eta function, which is similar to the zeta function. The eta function's pole and trivial zeros are the same as those of the zeta function. Furthermore, the Chi and the eta functions have the same non-trivial zeros on the imaginary axis. Here, the imaginary axis corresponds to the critical line of the eta function. And I proposed a generalized Riemann hypothesis for the eta function that states that all non-trivial zeros lie on the imaginary axis. Since I was able to discover the explicit formula for the eta function, the deductive method was used to prove the generalized Riemann hypothesis for the eta function.
Category: Number Theory
[10] viXra:2307.0136 [pdf] replaced on 2025-02-12 05:24:05
Authors: Olivier Massot
Comments: Clearer writing of subsections 2.6.1 and 2.6.2 pages 27 & 28
This study (in English) leads to a different formulation of the Newton's Binomial expansion. From there, a proof of Fermat's conjecture seems possible.
Category: Number Theory
[9] viXra:2307.0133 [pdf] submitted on 2023-07-25 21:20:01
Authors: Ahmed Idrissi Bouyahyaoui
Comments: 6 Pages. In English and French
Let xi = 2^αi*yi and vi = 2^βi*zi , x0, yi and zi are odd integers. The sequence {xi + vi} built by Collatz algorithm is a Collatz sequence if it exists n such that xn + vn = 1. By hypothesis S(y0) is a Collatz sequence, then it exists at least one i such that yi = 1, xi = 2^αi*yi = 2^αi and vi = zi (because vi < xi and xi + vi > 0). As for every k ≥ i yk Є [1, 4, 2], xk + vk is of form : xk + vk = 2^αk + zk. For every optimal point (k, xk + vk), continuous and differentiable function f(α) = x + v = 2^α + z has a zero derivative and the primitive function z = - 2^α + c, c is an arbitrary integer constant.For every optimum we have : f(α) = c. At the optimum minimum = 1, it exists at least one n such that, yn Є [1, 4, 2], f(αn) = xn + vn = 2^αn + zn = 2^αn - 2^αn + c = c. For the minimum f(αn) = 1, it suffices to set c = 1 and so we have :f(αn) = xn + vn = 1, xn = 2^αn and vn = — (2^αn — 1). Conclusion : The sequence S(x0 +2) ends in 1 and has the only cycle [1, 4, 2, 1].So by recurrence, every positive integer number gives a Collatz sequence.
Category: Number Theory
[8] viXra:2307.0129 [pdf] submitted on 2023-07-24 00:20:07
Authors: Radomir Majkic
Comments: 11 Pages.
If Legendre conjecture does not hold all integers in the interior of BT (n^{2},(n+1)^{2})ET are composed numbers. The composite integers counting shown that the rate of the number of the odd composites to the number of odd integers in theinterior of BT (n^{2},(n+1)^{2})ET is smaller than one. Consequently, the Legendre conjecture holds.
Category: Number Theory
[7] viXra:2307.0127 [pdf] submitted on 2023-07-24 05:28:17
Authors: Rakesh Timsina
Comments: 9 Pages.
This paper presents a geometrical approach to tackle the infamous Collatz conjecture. In this approach, we represent odd natural numbers as points in 2-D space. We then define a iterative geometrical algorithm and prove that this algorithm is equivalent to the Collatz function (more precisely, Syracuse function). Using the monotone convergence theorem, we prove the sequence generated by this algorithm always converges to 1. Since, this is same as saying Collatz (Syracuse) sequence converges to 1, we prove that the Collatz conjecture is true.
Category: Number Theory
[6] viXra:2307.0126 [pdf] replaced on 2025-02-12 05:35:22
Authors: Olivier Massot
Comments: Clearer writing of subsections 2.6.1 and 2.6.2 pages 27 & 28
This study (in French) leads to a different formulation of the Newton's Binomial expansion. From there, a proof of Fermat's conjecture seems possible.
Category: Number Theory
[5] viXra:2307.0122 [pdf] submitted on 2023-07-23 22:43:53
Authors: Giovanni Di Savino
Comments: 1 Page.
It is mathematical that: there are more planes in the ocean than submarines in the sky; there are more unaccepted essays than published essays; referring to the correspondence used to control the herd 3500 years ago, the mathematical problem defined as the oldest of mathematical problems "Why cannot exist perfect odd numbers" is mathematically demonstrated, but it is with mathematics, applied to the herd, that it is demonstrated that there are even perfect numbers and there are no perfect odd solutions.
Category: Number Theory
[4] viXra:2307.0116 [pdf] submitted on 2023-07-22 08:36:18
Authors: Yinghao Luo
Comments: 3 Pages.
The Riemann zeta function in the Riemann hypothesis equals zero for both all negative even integers and an infinite number of complex numbers with real part 1/2. We can conclude that Riemann hypothesis is undecidable.
Category: Number Theory
[3] viXra:2307.0092 [pdf] replaced on 2023-08-08 22:33:02
Authors: Oussama Basta
Comments: 2 Pages. A better version, Email included, submitted to a journal
This paper presents a rigorous proof of the Erdős-Straus conjecture, demonstrating that the equation (w^2−F1F2) / F^2 = 1 holds true for all n ≥ 2. The Erdős-Straus conjecture, originally formulated by mathematicians Paul Erdős and Ernst G. Straus, relates to the representation of positive integers as the sum of three reciprocal fractions. Through a series of mathematical derivations and substitution, we prove the conjecture and provide insight into its implications.
Category: Number Theory
[2] viXra:2307.0086 [pdf] replaced on 2024-03-19 19:58:52
Authors: Oussama Basta
Comments: 11 Pages. More detailed and has Latex Equation and fixed spelling mistakes
The Riemann Hypothesis, a famous unsolved problem in mathematics, posits a deep connection between the distribution of prime numbers and the nontrivial zeros of the Riemann zeta function. In this study, we investigate the presence of zeros at prime numbers in a specific mathematical expression, $ln (sec (pi cdot nlog(n)))$, and its implications for the Riemann hypothesis. By employing rigorous mathematical analysis, we establish a clear connection between prime numbers, trigonometric functions, and the behavior of the Riemann zeta function. Our findings contribute to the body of knowledge surrounding the Riemann hypothesis and its potential proof, shedding light on the intricate nature of prime numbers and their relationship to fundamental mathematical functions.
Category: Number Theory
[1] viXra:2307.0018 [pdf] submitted on 2023-07-04 17:22:29
Authors: Giovanni Di Savino
Comments: 2 Pages.
A perfect number is a natural number which is equal to the sum of its integer divisors including 1 and excluding itself, but a number n is also perfect in which the sum of its divisors including 1 and itself is equal to 2n. The natural numbers are infinite, for each of them there is a successor number and if it will never be possible to know how many, among the natural numbers, there can be perfect numbers, it is possible to know why there are even perfect numbers and there cannot be odd perfect numbers .The perfect number equal to 2n recalls a measurement technique, used 35,000 years ago when numbers were not known and which is similar to today's one-to-one correspondence. The correspondence of years ago consisted in associating each element of a set A with an element of set B; a concrete correspondence today is: "in a shirt the A.soles can be associated with the B.brass". Years ago, not knowing how to count, any set A was made to correspond to a set B in order to obtain that any difference between the two sets was the confirmation or not that the two sets were equal.
Category: Number Theory
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