scholarly journals Prime Numbers: A Particle in a Box and the Complex Wave Model

2015 ◽  
Vol 7 (4) ◽  
pp. 43
Author(s):  
Raul Alberto Ribeiro Correia de Sousa
Keyword(s):  
Prime Number ◽  
Energy Levels ◽  
Computing Time ◽  
Wave Model ◽  
Prime Numbers ◽  
Integer Number ◽  
Oscillation Modes ◽  
Complex Wave ◽  
New Formula ◽  
The Relationship

<p>Euler{'}s formula establishes the relationship between the trigonometric function and the exponential function. In doing so unifies two waves, a real and an imaginary one, that propagate through the Complex number set, establishing relation between integer numbers. A complex wave, if anchored by zero and by a defined integer number \textit{N}, only can assume certain oscillation modes. The first mode of oscillation corresponds always to a \textit{N} prime number and the other modes to its multiples.</p><p>\begin{center}<br />\(\psi (x)=x e^{i\left(\frac{n \pi }{N}x\right)}\)<br />\end{center}</p><p>Under the above described conditions, these waves and their admissible oscillation modes allows for primality testing of integer numbers, the deduction of a new formula $\pi(x)$ for counting prime numbers and the identification of patterns in the prime numbers distribution with computing time gains in the calculations. In this article, four theorems and one rule of factorizing are put forward with consequences for prime number signaling, counting and distribution. Furthermore, it is establish the relationship between this complex wave with a time independent semi-classical harmonic oscillator, in which the spectrum of the allowed energy levels are always only prime numbers. Thus, it is affirmative the reply to the question if the prime numbers distribution is related to the energy levels of a physical system.</p>

scholarly journals Identification of Composite Combinations: Key to Validate Goldbach Conjecture

2020 ◽  
Author(s):  
Manish Khare ◽  
Kalyanlakshmi Chitta
Keyword(s):  
Prime Number ◽  
Prime Numbers ◽  
Integer Number ◽  
Multiplicative Property ◽  
General Terms ◽  
Successive Addition ◽  
Detailed Method ◽  
Cyclical Pattern ◽  
General Method

This paper discusses a possible approach to validate the Goldbach conjucture which states that all even numbers can be expressed as a summation of two prime numbers. For this purpose the paper begins with the concept of successive-addition-of-digits-of-an-integer-number (SADN) and its properties in terms of basic algebraic functions like addition, multiplication and subtraction. This concept of SADN forms the basis for classifying all odd numbers into 3 series- the S1, S3 and S5 series- which comprise of odd numbers of SADN(7,4,1), SADN(3,9,6) and SADN(5,2,8) respectively and follow a cyclical order. The S1 and S5 series are of interest in the analysis since they include both prime and composite numbers while the S3 series exclusively consists of composite numbers. Furthermore, the multiplicative property of SADN shows why composites on the S1 series are derived as products of intra-series elements of the S1 and S5 series while composites on the S5 series are derived as products of inter-series elements of the S1 and S5 series. The role of SADN is also important in determining the relevant series for identifying the combination of primes for a given even number since it shows why such combinations for even numbers of SADN(1,4,7) will be found on the S5 series while those for even numbers of SADN(2,5,8) will lie on the S1 series and both the series have a role to play in identifying the prime number combinations for even numbers with SADN(3,6,9). Thereafter, the analysis moves to calculating the total number of acceptable combinations for a given even number that would include combinations in the nature of two composites (c1+c2), one prime and one composite (p+c) and two primes (p1+p2). A cyclical pattern followed by even numbers is also discussed in this context. Identifying the c1+c2 and p+c combinations and thereafter subtracting them from the total number of combinations will yield the number of p1+p2 combinations. For this purpose the paper discusses a general method to calculate the number of composites on the S1 and S5 series for a given number and provides a detailed method for deriving the number of c1+c2 combinations. The paper presents this analysis as a proof to validate the Goldbach conjecture. Since even numbers can be of SADN 1 to 9 and the relation between nTc (i.e. total number of acceptable combinations) and nc(i.e. number of composites) for all even numbers can either be of nTc &gt; nc or nTc &le; nc, the paper shows that the Goldbach conjecture is true for both these categories of even numbers. In this manner this analysis is totally inclusive of all even numbers in general terms and since the analysis of every even number is common in methodology but unique in compilation, apart from being totally inclusive, it is also mutually exclusive in nature. This proves that the Goldbach conjecture which states that all even numbers can be expressed as atleast one combination of two prime numbers holds true for all even numbers, across all categories possible. Additionally this approach proves that the identification of p1+p2 combinations which would validate the Goldbach conjecture lies in the identification of c1+c2 combinations.

scholarly journals Identification of Composite Combinations: Key to Validate Goldbach Conjecture

2020 ◽  
Author(s):  
Manish Khare ◽  
Kalyanlakshmi Chitta
Keyword(s):  
Prime Number ◽  
Prime Numbers ◽  
Integer Number ◽  
Multiplicative Property ◽  
General Terms ◽  
Successive Addition ◽  
Detailed Method ◽  
Cyclical Pattern ◽  
General Method

This paper discusses a possible approach to validate the Goldbach conjucture which states that all even numbers can be expressed as a summation of two prime numbers. For this purpose the paper begins with the concept of successive-addition-of-digits-of-an-integer-number (SADN) and its properties in terms of basic algebraic functions like addition, multiplication and subtraction. This concept of SADN forms the basis for classifying all odd numbers into 3 series- the S1, S3 and S5 series- which comprise of odd numbers of SADN(7,4,1), SADN(3,9,6) and SADN(5,2,8) respectively and follow a cyclical order. The S1 and S5 series are of interest in the analysis since they include both prime and composite numbers while the S3 series exclusively consists of composite numbers. Furthermore, the multiplicative property of SADN shows why composites on the S1 series are derived as products of intra-series elements of the S1 and S5 series while composites on the S5 series are derived as products of inter-series elements of the S1 and S5 series. The role of SADN is also important in determining the relevant series for identifying the combination of primes for a given even number since it shows why such combinations for even numbers of SADN(1,4,7) will be found on the S5 series while those for even numbers of SADN(2,5,8) will lie on the S1 series and both the series have a role to play in identifying the prime number combinations for even numbers with SADN(3,6,9). Thereafter, the analysis moves to calculating the total number of acceptable combinations for a given even number that would include combinations in the nature of two composites (c1+c2), one prime and one composite (p+c) and two primes (p1+p2). A cyclical pattern followed by even numbers is also discussed in this context. Identifying the c1+c2 and p+c combinations and thereafter subtracting them from the total number of combinations will yield the number of p1+p2 combinations. For this purpose the paper discusses a general method to calculate the number of composites on the S1 and S5 series for a given number and provides a detailed method for deriving the number of c1+c2 combinations. The paper presents this analysis as a proof to validate the Goldbach conjecture. Since even numbers can be of SADN 1 to 9 and the relation between nTc (i.e. total number of acceptable combinations) and nc(i.e. number of composites) for all even numbers can either be of nTc &gt; nc or nTc &le; nc, the paper shows that the Goldbach conjecture is true for both these categories of even numbers. In this manner this analysis is totally inclusive of all even numbers in general terms and since the analysis of every even number is common in methodology but unique in compilation, apart from being totally inclusive, it is also mutually exclusive in nature. This proves that the Goldbach conjecture which states that all even numbers can be expressed as atleast one combination of two prime numbers holds true for all even numbers, across all categories possible. Additionally this approach proves that the identification of p1+p2 combinations which would validate the Goldbach conjecture lies in the identification of c1+c2 combinations.

scholarly journals Energy-Constrained Design of Joint NOMA-Diversity Schemes with Imperfect Interference Cancellation

Sensors ◽  
2021 ◽  
Vol 21 (12) ◽  
pp. 4194
Author(s):  
Fulvio Babich ◽  
Giulia Buttazzoni ◽  
Francesca Vatta ◽  
Massimiliano Comisso
Keyword(s):  
Interference Cancellation ◽  
Energy Levels ◽  
Random Access ◽  
Actual Performance ◽  
Available Energy ◽  
Fifth Generation ◽  
Code Modulation ◽  
Energy Constrained ◽  
The Relationship ◽  
The Internet Of Things

This study proposes a set of novel random access protocols combining Packet Repetition (PR) schemes, such as Contention Resolution Diversity Slotted Aloha (CRDSA) and Irregular Repetition SA (IRSA), with Non Orthogonal Multiple Access (NOMA). Differently from previous NOMA/CRDSA and NOMA/IRSA proposals, this work analytically derives the energy levels considering two realistic elements: the residual interference due to imperfect Interference Cancellation (IC), and the presence of requirements on the power spent for the transmission. More precisely, the energy-limited scenario is based on the relationship between the average available energy and the selected code modulation pair, thus being of specific interest for the implementation of the Internet of Things (IoT) technology in forthcoming fifth-generation (5G) systems. Moreover, a theoretical model based on the density evolution method is developed and numerically validated by extensive simulations to evaluate the limiting throughput and to explore the actual performance of different NOMA/PR schemes in energy-constrained scenarios.

scholarly journals The role of L/W in the solar radiation for Saharian cities

2019 ◽  
Vol 91 ◽  
pp. 05006
Author(s):  
Rami Qaoud ◽  
Alkama Djamal
Keyword(s):  
Solar Radiation ◽  
Solar Energy ◽  
Energy Levels ◽  
Direct Solar Radiation ◽  
Test Field ◽  
Natural Lighting ◽  
The Difference ◽  
The Impact ◽  
The Relationship

The urban fabric of the desert cities is based on the principle of reducing the impact of urban canyons on direct solar radiation. Here comes this research, which is based on a comparative study of the periods of direct solarisation and values of the solar energy of urban canyons via two urban fabrics that have different building densities, where the ratio between L/W is different. In order to obtain the real values of the solar energy (thermal, lighting), the test field was examined every two hours, each three consecutive days. The measurement stations are positioned by the three types of the relationship between L/W, (L≥2w, L=w, L≤0.5w). According to the results, we noticed and recorded the difference in the periods of direct solarization between the types of urban engineering canyons, reaching 6 hours a day, the difference in thermal values of air, reaching 4 °C, and the difference in periods of direct natural lighting, reaching 6 hours. It should be noted that the role of the relationship between L/W is to protect the urban canyons by reducing the impact of direct solar radiation on urban canyons, providing longer hours of shading, and reducing solar energy levels (thermal, lighting) at the urban canyons. This research is classified under the research axis (the studies of external spaces in the urban environment according to the bioclimatic approach and geographic approach). But this research aims to focus on the tracking and studying the distribution of the solar radiation - thermal radiation and lighting radiation - in different types of street canyons by comparing the study of the direct solarization periods of each type and the quantity of solar energy collected during the solarization periods.

scholarly journals The continuity of prime numbers can lead to even continuity(The ultimate)

2021 ◽  
Author(s):  
Xie Ling
Keyword(s):  
Prime Number ◽  
Prime Numbers ◽  
Original Group

Abstract n continuous prime numbers can combine a group of continuous even numbers. If an adjacent prime number is followed, the even number will continue. For example, if we take prime number 3, we can get even number 6. If we follow an adjacent prime number 5, we can get even numbers by using 3 and 5: 6, 8 and 10. If a group of continuous prime numbers 3,5,7,11,... P, we can get a group of continuous even numbers 6,8,10,12,..., 2n. Then if an adjacent prime number q is followed, the original group of even numbers 6,8,10,12,..., 2n will be finitely extended to 2 (n + 1) or more adjacent even numbers. My purpose is to prove that the continuity of prime numbers will lead to even continuity as long as 2 (n + 1) can be extended. If the continuity of even numbers is discontinuous, it violates the Bertrand Chebyshev theorem of prime numbers. Because there are infinitely many prime numbers: 3, 5, 7, 11,... We can get infinitely many continuous even numbers: 6,8,10,12,...

scholarly journals Derivation Power Sums of Even Integer Number Formula

2013 ◽  
Vol 10 (2) ◽  
pp. 301-318
Author(s):  
Baghdad Science Journal
Keyword(s):  
Programming Language ◽  
Computer Programming ◽  
Integer Number ◽  
Power Sums ◽  
Derivative Method ◽  
Number Formula ◽  
Microsoft Office ◽  
The Relationship

This paper included derivative method for the even r power sums of even integer numbers formula to approach high even (r+2) power sums of even integer numbers formula so on we can approach from derivative odd r power sums of even integer numbers formula to high odd (r+2) power sums of even integer numbers formula this derivative excellence have ability to used by computer programming language or any application like Microsoft Office Excel. Also this research discovered the relationship between r power sums of even integer numbers formula and both formulas for same power sums of odd integer numbers formula and for r power sums of all integer numbers formula in another way.

scholarly journals Anisotropic Diffusion and the Townsend?Huxley Experiment

1973 ◽  
Vol 26 (4) ◽  
pp. 469 ◽  
Author(s):  
JJ Lowke
Keyword(s):  
Experimental Data ◽  
Electric Field ◽  
Electron Mobility ◽  
Diffusion Coefficients ◽  
Applied Electric Field ◽  
Anisotropic Diffusion ◽  
Diffusion Tubes ◽  
New Formula ◽  
The Relationship ◽  
Diffusion Tube

The relationship between current ratios and electron diffusion coefficients for the Townsend-Huxley experiment is reanalysed with the assumption that diffusion can be represented by two coefficients DT and DL for diffusion transverse and parallel respectively to the applied electric field. When the new formula is used to interpret previous experimental data obtained with a diffusion tube of length 2 cm, the derived values of DT/fl become independent of pressure (fl being the electron mobility). For longer diffusion tubes (~ 6 cm), current ratios are insensitive to DL and the results differ insignificantly from those obtained using the formula previously derived on the assumption that diffusion is isotropic.

scholarly journals The continuity of prime numbers can lead to even continuity(The ultimate)

2021 ◽  
Author(s):  
Xie Ling
Keyword(s):  
Prime Number ◽  
Prime Numbers ◽  
Original Group

Abstract n continuous prime numbers can combine a group of continuous even numbers. If an adjacent prime number is followed, the even number will continue. For example, if we take prime number 3, we can get even number 6. If we follow an adjacent prime number 5, we can get even numbers by using 3 and 5: 6, 8 and 10. If a group of continuous prime numbers 3,5,7,11,... P, we can get a group of continuous even numbers 6,8,10,12,..., 2n. Then if an adjacent prime number q is followed, the original group of even numbers 6,8,10,12,..., 2n will be finitely extended to 2 (n + 1) or more adjacent even numbers. My purpose is to prove that the continuity of prime numbers will lead to even continuity as long as 2 (n + 1) can be extended. If the continuity of even numbers is discontinuous, it violates the Bertrand Chebyshev theorem of prime numbers. Because there are infinitely many prime numbers: 3, 5, 7, 11,... We can get infinitely many continuous even numbers: 6,8,10,12,...

scholarly journals RSA algorithm using key generator ESRKGS to encrypt chat messages with TCP/IP protocol

2020 ◽  
Vol 8 (2) ◽  
pp. 113-120
Author(s):  
Aminudin Aminudin ◽  
Gadhing Putra Aditya ◽  
Sofyan Arifianto
Keyword(s):  
Prime Number ◽  
Instant Messaging ◽  
Prime Numbers ◽  
Public Key ◽  
Key Generation ◽  
Security Testing ◽  
Rsa Algorithm ◽  
Computer Resource ◽  
Encryption And Decryption ◽  
Private And Public

This study aims to analyze the performance and security of the RSA algorithm in combination with the key generation method of enhanced and secured RSA key generation scheme (ESRKGS). ESRKGS is an improvement of the RSA improvisation by adding four prime numbers in the property embedded in key generation. This method was applied to instant messaging using TCP sockets. The ESRKGS+RSA algorithm was designed using standard RSA development by modified the private and public key pairs. Thus, the modification was expected to make it more challenging to factorize a large number n into prime numbers. The ESRKGS+RSA method required 10.437 ms faster than the improvised RSA that uses the same four prime numbers in conducting key generation processes at 1024-bit prime number. It also applies to the encryption and decryption process. In the security testing using Fermat Factorization on a 32-bit key, no prime number factor was found. The test was processed for 15 hours until the test computer resource runs out.

scholarly journals Exploiting the Relationship between Pruning Ratio and Compression Effect for Neural Network Model Based on TensorFlow

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Bo Liu ◽  
Qilin Wu ◽  
Yiwen Zhang ◽  
Qian Cao
Keyword(s):  
Neural Network ◽  
Network Model ◽  
Neural Network Model ◽  
Prediction Accuracy ◽  
Computing Time ◽  
Experimental Results ◽  
Data Set ◽  
Compression Effect ◽  
Model Based ◽  
The Relationship

Pruning is a method of compressing the size of a neural network model, which affects the accuracy and computing time when the model makes a prediction. In this paper, the hypothesis that the pruning proportion is positively correlated with the compression scale of the model but not with the prediction accuracy and calculation time is put forward. For testing the hypothesis, a group of experiments are designed, and MNIST is used as the data set to train a neural network model based on TensorFlow. Based on this model, pruning experiments are carried out to investigate the relationship between pruning proportion and compression effect. For comparison, six different pruning proportions are set, and the experimental results confirm the above hypothesis.

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