prime numbers
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2022 ◽  
Vol 14 (1) ◽  
pp. 55
Author(s):  
Shaimaa said soltan

In this document, we will present a new way to visualize the distribution of Prime Numbers in the number system to spot Prime numbers in a subset of numbers using a simpler algorithm. Then we will look throw a classification algorithm to check if a number is prime using only 7 simple arithmetic operations with an algorithm complexity less than or equal to O (7) operations for any number.


2022 ◽  
Author(s):  
Xie Ling
Keyword(s):  

Abstract Two prime numbers{3,5}.{3,5}→{3+3= 6,3+5=8,5+5=10} →{6,8,10}.{{10}→(5+5=10 = 3 + 7) → 7}Increased by 7 →{3, 5, 7}.{3,5,7}→{ 3+3= 6, 3+5=8, 5+5=10, 5+7=12, 7+7=14}→{ 6,8,10,12,14}.{{14}→(7+7=14 = 3 + 11) → 11} Increased by 11 →{3, 5, 7, 11}.Note changes: {3,5,7}→{ 6,8,10,12,14}→ {3, 5, 7,11}.The same logic would be:{3,5,7,11}→{ 6,8,10,12,14,16}→ {3, 5, 7,11,13}.The same logic would be:{3,5,7,11,13}→{ 6,8,10,12,14,16,18,20}→ {3, 5, 7,11,13,17}.If you expand infinitely in the above specified mode: {3, 5, 7, 11, 13, 17 ,...Get: { 6,8,10,12,14,16,18,20,22,...The above is: the continuity of prime numbers can lead to even continuity.Get: Goldbach conjecture holds.If it is mandatory: Authenticity stops at an even number 2n.{{3, 5, 7, 11 ,...,p1}→{ 6,8,10,12,...,2n}.{3, 5, 7, 11, ...,p1} ↛(2n+2).∀p+∀p≠2n+2 } It can be proved that: It violates the "Bertrand Chebyshev" theorem.∴ {3, 5, 7, 11, 13, ...→{ 6,8,10,12,14,...Get: Goldbach conjecture holds.


Quantum ◽  
2022 ◽  
Vol 6 ◽  
pp. 614
Author(s):  
Honghao Fu

Let p be an odd prime and let r be the smallest generator of the multiplicative group Zp∗. We show that there exists a correlation of size Θ(r2) that self-tests a maximally entangled state of local dimension p−1. The construction of the correlation uses the embedding procedure proposed by Slofstra (Forum of Mathematics, Pi. (2019)). Since there are infinitely many prime numbers whose smallest multiplicative generator is in the set {2,3,5} (D.R. Heath-Brown The Quarterly Journal of Mathematics (1986) and M. Murty The Mathematical Intelligencer (1988)), our result implies that constant-sized correlations are sufficient for self-testing of maximally entangled states with unbounded local dimension.


Entropy ◽  
2021 ◽  
Vol 24 (1) ◽  
pp. 60
Author(s):  
Ernesto P. Borges ◽  
Takeshi Kodama ◽  
Constantino Tsallis

The rich history of prime numbers includes great names such as Euclid, who first analytically studied the prime numbers and proved that there is an infinite number of them, Euler, who introduced the function ζ(s)≡∑n=1∞n−s=∏pprime11−p−s, Gauss, who estimated the rate at which prime numbers increase, and Riemann, who extended ζ(s) to the complex plane z and conjectured that all nontrivial zeros are in the R(z)=1/2 axis. The nonadditive entropy Sq=k∑ipilnq(1/pi)(q∈R;S1=SBG≡−k∑ipilnpi, where BG stands for Boltzmann-Gibbs) on which nonextensive statistical mechanics is based, involves the function lnqz≡z1−q−11−q(ln1z=lnz). It is already known that this function paves the way for the emergence of a q-generalized algebra, using q-numbers defined as ⟨x⟩q≡elnqx, which recover the number x for q=1. The q-prime numbers are then defined as the q-natural numbers ⟨n⟩q≡elnqn(n=1,2,3,⋯), where n is a prime number p=2,3,5,7,⋯ We show that, for any value of q, infinitely many q-prime numbers exist; for q≤1 they diverge for increasing prime number, whereas they converge for q>1; the standard prime numbers are recovered for q=1. For q≤1, we generalize the ζ(s) function as follows: ζq(s)≡⟨ζ(s)⟩q (s∈R). We show that this function appears to diverge at s=1+0, ∀q. Also, we alternatively define, for q≤1, ζqΣ(s)≡∑n=1∞1⟨n⟩qs=1+1⟨2⟩qs+⋯ and ζqΠ(s)≡∏pprime11−⟨p⟩q−s=11−⟨2⟩q−s11−⟨3⟩q−s11−⟨5⟩q−s⋯, which, for q<1, generically satisfy ζqΣ(s)<ζqΠ(s), in variance with the q=1 case, where of course ζ1Σ(s)=ζ1Π(s).


Author(s):  
Martin Lind

We consider an equidistributed concatenation sequence of pseudorandom rational numbers generated from the primes by an inversive congruential method. In particular, we determine the sharp convergence rate for the star discrepancy of said sequence. Our arguments are based on well-known discrepancy estimates for inversive congruential pseudorandom numbers together with asymptotic formulae involving prime numbers.


Author(s):  
Ahmet Furkan Gocgen

Integer is either a composite number or a prime number. Therefore, detecting composite numbers is important for solving prime numbers. The study of prime numbers, apart from satisfying human curiosity, can be very important. In this article, the order of composite numbers has been detected. And explained with a simple method and a simple function. And, a method has been developed in which all composite numbers and therefore prime numbers can be determined by using the specified methods, functions and formulas.


2021 ◽  
Author(s):  
Fabrizio Tamburini ◽  
Ignazio Licata

Abstract The Riemann Hypothesis states that the Riemann zeta function ζ(z) admits a set of “non-trivial” zeros that are complex numbers supposed to have real part 1/2. Their distribution on the complex plane is thought to be the key to determine the number of prime numbers before a given number. Hilbert and Pólya suggested that the Riemann Hypothesis could be solved through the mathematical tools of physics, finding a suitable Hermitian or unitary operator that describe classical or quantum systems, whose eigenvalues distribute like the zeros of ζ(z). A different approach is that of finding a correspondence between the distribution of the ζ(z) zeros and the poles of the scattering matrix S of a physical system. Our contribution is articulated in two parts: in the first we apply the infinite-components Majorana equation in a Rindler spacetime and compare the results with those obtained with a Dirac particle following the Hilbert-Pólya approach showing that the Majorana solution has a behavior similar to that of massless Dirac particles and finding a relationship between the zeros of zeta end the energy states. Then, we focus on the S-matrix approach describing the bosonic open string scattering for tachyonic states with the Majorana equation. Here we find that, thanks to the relationship between the angular momentum and energy/mass eigenvalues of the Majorana solution, one can explain the still unclear point for which the poles and zeros of the S-matrix of an ideal system that can satisfy the Riemann Hypothesis, exist always in pairs and are related via complex conjugation. As claimed in the literature, if this occurs and the claim is correct, then the Riemann Hypothesis could be in principle satisfied, tracing a route to a proof.


2021 ◽  
Vol 62 ◽  
pp. 80-85
Author(s):  
Edmundas Mazėtis ◽  
Grigorijus Melničenko

The authors introduced the concept of a pseudo-Heron triangle, such that squares of sides are integers, and the area is an integer multiplied by $2$. The article investigates the case of pseudo-Heron triangles such that the squares of the two sides of the pseudo-Heron triangle are primes of the form $4k+1$. It is proved that for any two predetermined prime numbers of the form $4k+1$ there exist pseudo-Heron triangles with vertices on an integer lattice, such that these two primes are the sides of these triangles and such triangles have a finite number. It is also proved that for any predetermined prime number of the form $4k+1$, there are isosceles triangles with vertices on an integer lattice, such that this prime is equal to the values of two sides and there are only a finite number of such triangles.


Author(s):  
Ahmet Furkan Gocgen

Integer is either a composite number or a prime number. Therefore, detecting composite numbers is important for solving prime numbers. The study of prime numbers, apart from satisfying human curiosity, can be very important. In this article, the order of composite numbers has been detected. And explained with a simple method and a simple function. And, a method has been developed in which all composite numbers and therefore prime numbers can be determined by using the specified methods, functions and formulas.


2021 ◽  
Author(s):  
Michaela Cully-Hugill ◽  
Ethan Lee

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