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[53] viXra:2003.0656 [pdf] submitted on 2020-03-30 02:25:08
Authors: Shekhar Suman
Comments: 5 Pages.
Riemann zeta
Category: Number Theory
[52] viXra:2003.0655 [pdf] submitted on 2020-03-30 03:19:31
Authors: Yuji Masuda
Comments: 1 Page.
This is a break time story.
Category: Number Theory
[51] viXra:2003.0634 [pdf] submitted on 2020-03-29 04:20:11
Authors: Shekhar Suman
Comments: 8 Pages.
Riemann zeta function, completed riemann zeta function
Category: Number Theory
[50] viXra:2003.0630 [pdf] submitted on 2020-03-29 06:36:29
Authors: Odoardo Volonterio, Michele Nardelli
Comments: 56 Pages.
In this work we have described a new mathematical application concerning the discrete and the analytic functions: the Volonterio’s Transform (V Transform) and the Volonterio’s Polynomial. We have described various mathematical applications and properties of them, precisely the series development of the type Nk+M. Furthermore, we have showed also various examples and the possible mathematical connections with some sectors of Number Theory and String Theory.
Category: Number Theory
[49] viXra:2003.0629 [pdf] submitted on 2020-03-29 06:41:30
Authors: Odoardo Volonterio, Michele Nardelli, Francesco Di Noto
Comments: 55 Pages.
In this work we have described a new mathematical application concerning the discrete and the analytic functions: the Volonterio’s Transform and the Volonterio’s Polynomial. The Volonterio’s Transform (V Transform), indeed, work from the
world of discrete functions to the world of analytic functions. We have described various mathematical applications and properties of them. Furthermore, we have showed also various examples and the possible mathematical connections with some sectors of Number Theory and String Theory.
Category: Number Theory
[48] viXra:2003.0621 [pdf] submitted on 2020-03-28 11:38:43
Authors: Pedro Jesus Caceres
Comments: 13 Pages.
In this paper, we define the C-transformation as:
C_n {f}= ∑(k=1,n) f(k)- ∫f(n)dn(1)
And the C-values as:
C{f}=lim(n→∞)C_n {f}(2)
And we obtain a new representation for ζ(z) in the form ζ(z) = X(z) – Y(z) applying the C-transformation to the function f(x)=1/x^z for z∈C,Re(z)≥0,z≠1.
Category: Number Theory
[47] viXra:2003.0589 [pdf] submitted on 2020-03-27 09:59:38
Authors: Shekhar Suman
Comments: 6 Pages.
The Riemann Zeta function is defined as \\\\
\zeta(s)= $$\sum_{n=1}^{\infty} 1/n^{s}$$ ,\ Re(s)$$>$$1 \\\\
In \ this \ article \ we \ prove \ an \ equivalent \ of \ the \ Riemann \ Hypothesis.
Category: Number Theory
[46] viXra:2003.0585 [pdf] submitted on 2020-03-27 11:32:32
Authors: Michele Nardelli, Antonio Nardelli
Comments: 57 Pages.
In this paper we have described some Ramanujan’s integral equations and incomplete elliptic integrals of the first kind. Furthermore, we describe new possible mathematical connections with ϕ, ζ(2), and various parameters of Particle Physics
Category: Number Theory
[45] viXra:2003.0580 [pdf] submitted on 2020-03-26 14:59:22
Authors: Michele Nardelli, Antonio Nardelli
Comments: 90 Pages.
In this paper we have described some Ramanujan’s integrals of theta-functions and incomplete elliptic integrals of the first kind. Furthermore, we describe new possible mathematical connections with ϕ, ζ(2), and various parameters of Particle Physics
Category: Number Theory
[44] viXra:2003.0576 [pdf] submitted on 2020-03-26 01:42:33
Authors: Shekhar Suman
Comments: 3 Pages.
Riemann Hypothesis
Category: Number Theory
[43] viXra:2003.0571 [pdf] submitted on 2020-03-26 05:37:15
Authors: Andrea Berdondini
Comments: 4 Pages.
ABSTRACT. In this article we present a procedure for the determination of the upper bounds for prime gaps different from the most famous and known approaches. The proposed method analyzes the distribution of prime numbers using the set of relative integers ℤ. Using negative numbers too, it becomes intuitive to understand that that the arrangement of 2P+1 consecutive numbers that goes -P to P, is the only arrangement that minimizes the distance between two powers having the same absolute value of the base D, with |��|≤��. This arrangement is considered important because by increasing the number of powers of the prime numbers within a range of consecutive numbers, it is presumed to decrease the overlap between the prime numbers considered. Therefore, by reducing these overlaps, we suppose to obtain an arrangement, in which the prime numbers less than and equal to P and their multiples occupy the greatest possible number of positions within a range of 2P+1 consecutive numbers. Consequently, the maximum gap between two consecutive prime numbers ����+1−���� can never be greater than 2����. If this result could be demonstrated, would imply the resolution of the Legendre’s conjecture.
Category: Number Theory
[42] viXra:2003.0560 [pdf] submitted on 2020-03-25 16:17:30
Authors: Michele Nardelli, Antonio Nardelli
Comments: 83 Pages.
In this paper we have described some Ramanujan’s differential equations: new possible mathematical connections with ϕ, ζ(2), and various parameters of Particle Physics
Category: Number Theory
[41] viXra:2003.0551 [pdf] replaced on 2020-06-30 23:01:32
Authors: Kaoru Motose
Comments: 2 Pages. Maybe the last replacement (7th): on 1 July 2020 in Japanease time.
We can see that Feit-Thompson conjecture is true using factorizations of cyclotomic polynomials on the finite prime fields.
Category: Number Theory
[40] viXra:2003.0494 [pdf] submitted on 2020-03-23 15:36:19
Authors: Theophilus Agama
Comments: 12 Pages.
In this paper we develop a general method for estimating correlations of the forms \begin{align}\sum \limits_{n\leq x}G(n)G(x-n),\nonumber
\end{align}and \begin{align}\sum \limits_{n\leq x}G(n)G(n+l)\nonumber
\end{align}for a fixed $1\leq l\leq x$ and where $G:\mathbb{N}\longrightarrow \mathbb{R}^{+}$. To distinguish between the two types of correlations, we call the first \textbf{type} $2$ correlation and the second \textbf{type} $1$ correlation. As an application we estimate the lower bound for the \textbf{type} $2$ correlation of the master function given by \begin{align}\sum \limits_{n\leq x}\Upsilon(n)\Upsilon(n+l_0)\geq (1+o(1))\frac{x}{2\mathcal{C}(l_0)}\log \log ^2x,\nonumber
\end{align}provided $\Upsilon(n)\Upsilon(n+l_0)>0$. We also use this method to provide a first proof of the twin prime conjecture by showing that \begin{align}\sum \limits_{n\leq x}\Lambda(n)\Lambda(n+2)\geq (1+o(1))\frac{x}{2\mathcal{C}(2)}\nonumber
\end{align}for some $\mathcal{C}:=\mathcal{C}(2)>0$.
Category: Number Theory
[39] viXra:2003.0478 [pdf] submitted on 2020-03-23 07:17:49
Authors: Michele Nardelli, Antonio Nardelli
Comments: 75 Pages.
In this paper we have described the parameters of SMBH 87 and some formulas concerning Primordial Black Holes in String Theory and Inflation. We described also new possible mathematical connections with some Ramanujan equations, ϕ, ζ(2) and Hausdorff dimension values
Category: Number Theory
[38] viXra:2003.0472 [pdf] submitted on 2020-03-22 11:35:56
Authors: Shekhar Suman
Comments: 4 Pages.
Riemann Xi function as hadamard product
Category: Number Theory
[37] viXra:2003.0441 [pdf] submitted on 2020-03-21 06:45:41
Authors: Michele Nardelli
Comments: 86 Pages. Introduction and summary in Italian
This paper is fundamentally a review, a thesis, of principal results obtained in some sectors of
Number Theory and String Theory of various authoritative theoretical physicists and
mathematicians. Precisely, we have described some mathematical results regarding the Fermat’s Last Theorem, the Mellin transform, the Riemann zeta function, the Ramanujan’s modular equations, how primes and adeles are related to the Riemann zeta functions and the p-adic and adelic string theory.
Furthermore, we show that also the fundamental relationship concerning the Palumbo-Nardelli
model (a general relationship that links bosonic string action and superstring action, i.e. bosonic and fermionic strings in all natural systems), can be related with some equations regarding the p-adic (adelic) string sector. Thence, in conclusion, we have described some new interesting connections that are been obtained between String Theory and Number Theory, with regard the arguments above mentioned.
Category: Number Theory
[36] viXra:2003.0438 [pdf] submitted on 2020-03-21 08:44:32
Authors: Shekhar Suman
Comments: 4 Pages.
Analytic continuation as Riemann Xi function
Category: Number Theory
[35] viXra:2003.0432 [pdf] submitted on 2020-03-21 11:23:41
Authors: Michele Nardelli
Comments: 91 Pages.
The present paper is a review, a thesis of some very important contributes of E. Witten, C. Beasley, R. Ricci, B. Basso et al. regarding various applications concerning the Jones polynomials, the Wilson loops and the cusp anomaly and integrability from string theory.
Category: Number Theory
[34] viXra:2003.0431 [pdf] submitted on 2020-03-21 11:38:33
Authors: Michele Nardelli
Comments: 56 Pages. Summary in Italian
In this paper we have described the mathematics concerning the String Theory, candidate Theory of Everything. In particular the Bosonic String and Superstring Actions, the Naked Singularity and Palumbo-Nardelli Model
Category: Number Theory
[33] viXra:2003.0430 [pdf] submitted on 2020-03-21 11:45:21
Authors: Michele Nardelli
Comments: 54 Pages.
The aim of this paper is that of show the further and possible connections between the p-adic and
adelic strings and Lagrangians with Riemann zeta function with some problems, equations and
theorems in Number Theory.
Category: Number Theory
[32] viXra:2003.0422 [pdf] submitted on 2020-03-20 09:33:21
Authors: Shekhar Suman
Comments: 5 Pages.
Riemann Zeta function
Category: Number Theory
[31] viXra:2003.0369 [pdf] submitted on 2020-03-18 09:19:03
Authors: Andrey B. Skrypnik
Comments: 27 Pages.
The Proof of the Beale Conjecture confirms the P versus NP problem
Category: Number Theory
[30] viXra:2003.0368 [pdf] submitted on 2020-03-18 09:21:47
Authors: Andrey B. Skrypnik
Comments: 15 Pages.
The Proof of the Beale Conjecture confirms the P versus NP problem
Category: Number Theory
[29] viXra:2003.0361 [pdf] submitted on 2020-03-17 12:21:17
Authors: Claude Henri Raymond Dequatre
Comments: 53 Pages.
A large number of systems of diophantine equations have been generated, paying particular attention to the solutions identified as prime numbers.
A statistical approach was taken to create such systems with random terms and coefficients. This paper gives a detailed view of the work carried out and the analysis of these solutions which are part of the set of prime numbers.
Data, observations and an analysis of results allowed to evaluate the frequency of prime number solutions, identify some structures in prime number solution scatter plots and some relationships between the number of prime solutions and the number of primes and finally establish two conjectures on the number of prime solutions generated by these linear diophantine equation systems.
Category: Number Theory
[28] viXra:2003.0354 [pdf] submitted on 2020-03-16 21:16:33
Authors: Pedro Hugo García Peláez
Comments: 5 Pages.
Numbers with prime factors a * b * c * d where a = 2 and b = 3
And they also fulfill that (a * d) - (b * c) = 1 or -1
Are triangular numbers.
Category: Number Theory
[27] viXra:2003.0335 [pdf] submitted on 2020-03-16 20:58:12
Authors: Pedro Hugo García Peláez
Comments: 5 Pages.
Los números con factores primos a*b*c*d donde a=2 y b=3
Y que Además cumplen que (a*d)-(b*c)= 1 o -1
Son números triangulares
Category: Number Theory
[26] viXra:2003.0323 [pdf] submitted on 2020-03-15 13:29:58
Authors: Michele Nardelli, Antonio Nardelli
Comments: 50 Pages.
In this paper we have described some Ramanujan Partition Congruences, and obtained several mathematical connections with ϕ, ζ(2) and various Fractal Hausdorff Dimensions values
Category: Number Theory
[25] viXra:2003.0303 [pdf] replaced on 2023-07-19 05:12:55
Authors: A. A. Frempong
Comments: 9 Pages. Copyright © by A. A. Frempong
An ingenious proof of Fermat's Last theorem has been covered in this paper. Fermat's Last theorem states that if A, B, C, n are positive integers; A, B, and C are coprime, and n > 2, then the equation A^n + B^n = C^n, has no solutions. The principles applied in the proof are based on the properties of the factored Beal equations. However, the proof is by contradiction. 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 greatest common power of the prime factors on the left side of the equation is the same as a power of the prime factor on the right side of the equation. High school students can learn and prove this theorem as a bonus question on a final class exam.
Category: Number Theory
[24] viXra:2003.0290 [pdf] submitted on 2020-03-14 12:46:58
Authors: Michele Nardelli, Antonio Nardelli
Comments: 86 Pages.
In this paper we have described some Asymptotic Formulas and Ramanujan Identities, and obtained several mathematical connections with ϕ, ζ(2) and various Fractal Hausdorff Dimensions values
Category: Number Theory
[23] viXra:2003.0284 [pdf] submitted on 2020-03-13 16:22:17
Authors: Michele Nardelli, Antonio Nardelli
Comments: 69 Pages.
In this paper we have described some Ramanujan equations and obtained some mathematical connections with ϕ and various expressions inherent the String Theory, in particular Open strings.
Category: Number Theory
[22] viXra:2003.0278 [pdf] submitted on 2020-03-13 21:09:14
Authors: Shekhar Suman
Comments: 4 Pages.
The proof involves the Riemann Xi function as a Hadamard product over its zeros.
Category: Number Theory
[21] viXra:2003.0274 [pdf] replaced on 2020-03-19 03:34:26
Authors: A. A. Frempong
Comments: 9 Pages. Copyright © by A. A. Frempong
In a page margin, the author proves directly 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. The principles applied in the proof are based on the properties of the factored Beal equation. 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 greatest common power of the prime factors on the left side of the equation is the same as a power of the prime factor on the right side of the equation.
Category: Number Theory
[20] viXra:2003.0239 [pdf] submitted on 2020-03-11 00:41:59
Authors: Pedro Jesus Caceres
Comments: 7 Pages.
In this paper, we will prove that the distribution of the nontrivial zeros of the Riemann Zeta function in the critical line (Re(z)=1/2) is not random. There is a relationship between the values of those zeros and the Harmonic function that leads to an algebraic relationship between any two zeros. We will also show a simple code to obtain zeros based on the Harmonic function.
Category: Number Theory
[19] viXra:2003.0238 [pdf] replaced on 2020-03-13 21:20:00
Authors: Pedro Jesus Caceres
Comments: 17 Pages.
The Riemann Zeta function or Euler–Riemann Zeta function, ζ(s), is a function of a complex variable z that analytically continues the sum of the Dirichlet series:
[1] ζ(z)=∑(k=1,∞) k^(-z)
The Riemann zeta function is a meromorphic function on the whole complex z-plane, which is holomorphic everywhere except for a simple pole at z = 1 with residue 1. One of the most important advance in the study of Prime numbers was the paper by Bernhard Riemann in November 1859 called “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse” (On the number of primes less than a given quantity). In this paper, Riemann gave a formula for the number of primes less than x in terms the integral of 1/log(x), and also provided insights into the roots (zeros) of the zeta function, formulating a conjecture about the location of the zeros of ζ(z) in the critical line Re(z)=1/2.
[2]Riemann Hypothesis: All nontrivial zeros lie on the critical line, or Re(z) = 1/2.
In this paper, we use the decomposition of the Riemann Zeta function in the form:
[3]ζ(z) = X(z) - Y(z)
Category: Number Theory
[18] viXra:2003.0189 [pdf] replaced on 2020-03-21 11:19:58
Authors: Pedro Jesus Caceres
Comments: 13 Pages.
In this paper, we define the C-transformation as:
[1]C_n {f}= ∑_(k=1,n) f(k)- ∫f(n) dn
And the C-values as:
[2]C{f}=lim(n→∞) C_n {f}
And we obtain a new representation for ζ(z) in the form ζ(z) = X(z) – Y(z) applying the C-transformation to the function f(x)=1/x^z for z∈C,Re(z)≥0,z≠1.
Category: Number Theory
[17] viXra:2003.0177 [pdf] submitted on 2020-03-09 09:42:38
Authors: Surajit Ghosh
Comments: 98 Pages.
Proof of riemann hypothesis.
Category: Number Theory
[16] viXra:2003.0171 [pdf] replaced on 2020-03-16 12:33:25
Authors: Nikolay Dementev
Comments: 9 Pages.
Calculation of probability of interrelation between all prime numbers is presented.
Category: Number Theory
[15] viXra:2003.0170 [pdf] submitted on 2020-03-08 15:29:20
Authors: Michele Nardelli, Antonio Nardelli
Comments: 75 Pages.
In this paper we have described some Ramanujan formulas and obtained some mathematical connections with ϕ and various equations concerning Modified Gravity Theory
Category: Number Theory
[14] viXra:2003.0155 [pdf] submitted on 2020-03-07 11:14:06
Authors: Shekhar Suman
Comments: 5 Pages.
The proof involves Hadamard Product representation of Riemann Xi function
Category: Number Theory
[13] viXra:2003.0148 [pdf] replaced on 2020-04-10 00:48:43
Authors: A. A. Frempong
Comments: 9 Pages. Copyright © by A. A. Frempong
On a single page, the author proves the original Beal conjecture, the equivalent Beal conjecture and Fermat's Last theorem. The original Beal conjecture states that if A^x + B^y = C^z,, where A, B, C, x, y, z are positive integers and x, y, x >2, then A, B, and C have a common prime factor. The equivalent Beal conjecture states that if A, B, C, x, y, z are positive integers and A, B, and C are coprime, and x, y, z >2,, then the equation A^x + B^y = C^z, has no solutions. Fermat's Last theorem states that if A, B, C, n are positive integers; A, B, and C are coprime, and n > 2, then the equation A^n + B^n = C^n,
has no solutions. The principles applied in the three proofs are based on the same properties of the factored Beal equation. However the proofs of the equivalent Beal conjecture and Fermat's last theorem are by contradiction. 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 greatest common power of the prime factors on the left side of the equation is the same as a power of the prime factor on the right side of the equation. High school students can learn and prove this conjecture as a bonus question on a final class exam.
Category: Number Theory
[12] viXra:2003.0090 [pdf] replaced on 2020-05-21 03:36:19
Authors: Espen Gaarder Haug
Comments: 4 Pages.
In this short note, we will quickly look at optimal binary number systems used in communication (or transactions) under the assumption that one must use energy to give away (send) numbers. We show that the current binary system is not the optimal binary number system as it can be arbitraged. We also show that there exist other optimal binary number systems in such a scenario. Naturally, one has to ask, ``Optimal for whom? -- For the one sending the number out, or for the one receiving the number?'' Alternatively, we can have a binary number system that, on average, is neutral for both sender and receiver. Numbers are typically only considered to have symbolic value, but if the money units were so small that they came in the smallest possibly energy units, then we could be forced to switch to a number system where the physical value of each number was equal to its symbolic value. That is to say, the physical value of three must be higher than the physical value of two, for example. Numbers are always physical because storing or sending a number from a computer requires bits, and bits of information require energy.
Category: Number Theory
[11] viXra:2003.0073 [pdf] submitted on 2020-03-03 13:16:57
Authors: Michele Nardelli, Antonio Nardelli
Comments: 63 Pages.
In this paper we have described some Ramanujan formulas and obtained some mathematical connections with ϕ and various equations concerning different sectors of Cosmology and Black Holes/Wormholes Physics.
Category: Number Theory
[10] viXra:2003.0070 [pdf] submitted on 2020-03-03 15:33:26
Authors: Michele Nardelli, Antonio Nardelli
Comments: 52 Pages.
In this paper we have described the possible mathematical connections between some Ramanujan’s equations and various formulas concerning several sectors of Theoretical Physics and Cosmology
Category: Number Theory
[9] viXra:2003.0066 [pdf] replaced on 2020-03-03 22:26:07
Authors: Michael Hatfield
Comments: 2 Pages.
Goldbach's conjecture is proven using the Chinese Remainder Theorem. It is shown that an even number 2N greater than four cannot exist if it is congruent to every prime p less than N (mod a different prime number).
Category: Number Theory
[8] viXra:2003.0055 [pdf] replaced on 2020-03-11 07:41:21
Authors: Juan Moreno Borrallo
Comments: 5 Pages.
In this brief paper it is proved that, for some positive integer n and some prime number q < n such that gcd (q,n) = 1, the number of coprime numbers to qn less than n is greater than phi(qn)/2q
Category: Number Theory
[7] viXra:2003.0050 [pdf] replaced on 2020-03-13 12:17:59
Authors: Ryan Zielinski
Comments: 8 Pages. Version 2 condenses the main result into a single case, discusses related work, and makes some minor changes. Both versions are licensed under the CC BY 4.0, a Creative Commons Attribution License.
In this paper we will prove a relationship for sums of powers of recursive integer sequences. Also, we will give a possible path to discovery. As corollaries of the main result we will derive relationships for familiar integer sequences like the Fibonacci, Lucas, and Pell numbers. Last, we will discuss some applications and point to further work.
Category: Number Theory
[6] viXra:2003.0029 [pdf] submitted on 2020-03-02 10:02:38
Authors: Michele Nardelli, Antonio Nardelli
Comments: 70 Pages.
In this paper we have described a Ramanujan formula and obtained some mathematical connections with the golden ratio and various equations concerning different sectors of Cosmology and Black Holes/Wormholes Physics.
Category: Number Theory
[5] viXra:2003.0027 [pdf] submitted on 2020-03-01 11:55:20
Authors: Andrea Berdondini
Comments: 10 Pages.
ABSTRACT. In this article we present a point of view that highlights the importance of finding the upper bounds for prime gaps, in order to solve the twin primes conjecture and the Goldbach’s conjecture. For this purpose, we present a procedure for the determination of the upper bounds for prime gaps different from the most famous and known approaches. The proposed method analyzes the distribution of prime numbers using the set of relative integers ℤ. Using negative numbers too, it becomes intuitive to understand that that the arrangement of 2P+1 consecutive numbers that goes -P to P, is the only arrangement that minimizes the distance between two powers having the same absolute value of the base D, with |��|≤��. This arrangement is considered important because by increasing the number of powers of the prime numbers within a range of consecutive numbers, it is presumed to decrease the overlap between the prime numbers considered. Consequently, by reducing these overlaps, we suppose to obtain an arrangement, in which the prime numbers less than and equal to P and their multiples occupy the greatest possible number of positions within a range of 2P+1 consecutive numbers. If this result could be demonstrated, would imply not only the resolution of the Legendre’s conjecture, but also a step forward in the resolution of the twin primes conjecture and the Goldbach’s conjecture.
Category: Number Theory
[4] viXra:2003.0025 [pdf] submitted on 2020-03-01 15:36:34
Authors: Yuji Masuda
Comments: 1 Page.
This is a fundamental proof of fibonacci sequence.
Category: Number Theory
[3] viXra:2003.0020 [pdf] submitted on 2020-03-01 01:29:14
Authors: A. A. Frempong
Comments: 11 Pages. Copyright © by A. A. Frempong
The author proves the original Beal conjecture, the equivalent Beal conjecture and Fermat's Last theorem, all on three pages. The original Beal conjecture states that if A^x + B^y = C^z, where A, B, C, x, y are positive integers and x, y, x > 2, then A, B, and C have a common prime factor. The equivalent Beal conjecture states that if A, B, C, x, y, z are positive integers and A, B, and C are coprime, with x, y, z >2, then the equation A^x + B^y = C^z has no solutions. Fermat's Last theorem states that if A, B, C, n are positive integers; A, B, and C are coprime, and, n >2, then the equation A^n + B^n = C^n has no solutions. The principles applied in the three proofs are based on the same properties of the factored Beal equation. However the proofs of the equivalent Beal conjecture and Fermat's Last theorem are by contradiction. 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 greatest common power of the prime factors on the left side of the equation is the same as a power of the prime factor on the right side of the equation. High school students can learn and prove this conjecture as a bonus question on a final class exam.
Category: Number Theory
[2] viXra:2003.0008 [pdf] replaced on 2024-03-19 05:59:00
Authors: Theophilus Agama
Comments: 8 Pages. This paper has been technically and substantially improved.
In this paper, using the method of compression, we recover the lower bound for the ErdH{o}s unit distance problem and provide an alternative proof to the distinct distance conjecture. In particular, in $mathbb{R}^k$ for all $kgeq 2$, we have begin{align}# bigg{||vec{x_j}-vec{x_t}||:~||vec{x_j}-vec{x_t}||=1,~1leq t,j leq n,~vec{x_j},~vec{x}_t in mathbb{R}^kbigg}gg_k frac{sqrt{k}}{2}n^{1+o(1)}.onumberend{align}We also show thatbegin{align}# bigg{d_j:d_j=||vec{x_s}-vec{y_t}||,~d_jeq d_i,~1leq s,tleq nbigg}gg_k frac{sqrt{k}}{2}n^{frac{2}{k}-o(1)}.onumber end{align}These lower bounds generalizes the lower bounds of the ErdH{o}s unit distance and the distinct distance problem to higher dimensions.
Category: Number Theory
[1] viXra:2003.0003 [pdf] submitted on 2020-03-01 07:50:44
Authors: Michele Nardelli, Antonio Nardelli
Comments: 54 Pages.
In this paper we have described several Ramanujan’s formulas and obtained some mathematical connections with various equations concerning different sectors of Theoretical Physics and Cosmology
Category: Number Theory
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