Prime Research Articles

Implementasi Penyisipan Pesan Teks Terenkripsi Menggunakan Kriptografi ElGamal pada Citra Digital Menggunakan Steganogarafi LSB

Message security is very important in maintaining the confidentiality of information, especially those containing important information whose contents should only be known by the entitled party. Therefore, security is needed for messages that are confidential so that messages cannot be intercepted by people who do not want them and the information content of the message becomes more secure. One way is to apply ElGamal and LSB cryptography methods to be able to provide security to text messages by inserting into digital images, the ElGamal algorithm focuses on the strength of the key in solving discrete logarithm problems. By utilizing large prime numbers and discrete logarithm problems which are quite difficult and also by applying the LSB method for text insertion, the security of the key is more guaranteed. The test results on five image objects with the .BMP extension on the application that was built were running well and correctly. The image that is inserted in the encryption process is the insertion of text in the image after it is decrypted and the image produced after the encryption and decryption process is no different from the original image and the same thing for the image size does not change

Linearity of the Position of Prime Numbers. The Order of Prime and Composite Numbers

In this paper it was shown that all prime numbers lie on 96 half-lines. At the same time, it was shown that if a given number does not lie on any of the above half-lines, then it is a composite number. A corresponding linear mathematical relationship was also derived and it was shown that all prime numbers must satisfy it. If a given number does not satisfy the above dependence, then it is a composite number. One linear equation and 8 numbers are enough to carry out all of the above-mentioned analyses. In addition, the astonishing cyclical nature of the occurrence of composite numbers in the set of natural numbers has been demonstrated.

Linearity of the Position of Prime Numbers. The Order of Prime and Composite Numbers

In this paper it was shown that all prime numbers lie on 96 half-lines. At the same time, it was shown that if a given number does not lie on any of the above half-lines, then it is a composite number. A corresponding linear mathematical relationship was also derived and it was shown that all prime numbers must satisfy it. If a given number does not satisfy the above dependence, then it is a composite number. One linear equation and 8 numbers are enough to carry out all of the above-mentioned analyses. In addition, the astonishing cyclical nature of the occurrence of composite numbers in the set of natural numbers has been demonstrated.

On the Jones polynomial modulo primes

AbstractWe derive an upper bound on the density of Jones polynomials of knots modulo a prime number $p$ , within a sufficiently large degree range: $4/p^7$ . As an application, we classify knot Jones polynomials modulo two of span up to eight.

A Quixotic Proof of Fermat’s Two Squares Theorem for Prime Numbers

Every odd prime number p has exactly different expressions as a sum ab + cd of two ordered products ab and cd such that . An easy corollary is a proof of Fermat’s Theorem expressing primes in as sums of two squares.

The prime number theorem for primes in arithmetic progressions at large values

Abstract Assuming the Riemann hypothesis, we prove the latest explicit version of the prime number theorem for short intervals. Using this result, and assuming the generalised Riemann hypothesis for Dirichlet L-functions is true, we then establish explicit formulae for $\psi(x,\chi)$, $\theta(x,\chi)$ and an explicit version of the prime number theorem for primes in arithmetic progressions that hold for general moduli $q\geq 3$. Finally, we restrict our attention to $q\leq 10\,000$ and use an exact computation to refine these results.

LSGP-USFNet: Automated Attention Deficit Hyperactivity Disorder Detection Using Locations of Sophie Germain's Primes on Ulam's Spiral-Based Features with Electroencephalogram Signals.

Anxiety, learning disabilities, and depression are the symptoms of attention deficit hyperactivity disorder (ADHD), an isogenous pattern of hyperactivity, impulsivity, and inattention. For the early diagnosis of ADHD, electroencephalogram (EEG) signals are widely used. However, the direct analysis of an EEG is highly challenging as it is time-consuming, nonlinear, and nonstationary in nature. Thus, in this paper, a novel approach (LSGP-USFNet) is developed based on the patterns obtained from Ulam's spiral and Sophia Germain's prime numbers. The EEG signals are initially filtered to remove the noise and segmented with a non-overlapping sliding window of a length of 512 samples. Then, a time-frequency analysis approach, namely continuous wavelet transform, is applied to each channel of the segmented EEG signal to interpret it in the time and frequency domain. The obtained time-frequency representation is saved as a time-frequency image, and a non-overlapping n × n sliding window is applied to this image for patch extraction. An n × n Ulam's spiral is localized on each patch, and the gray levels are acquired from this patch as features where Sophie Germain's primes are located in Ulam's spiral. All gray tones from all patches are concatenated to construct the features for ADHD and normal classes. A gray tone selection algorithm, namely ReliefF, is employed on the representative features to acquire the final most important gray tones. The support vector machine classifier is used with a 10-fold cross-validation criteria. Our proposed approach, LSGP-USFNet, was developed using a publicly available dataset and obtained an accuracy of 97.46% in detecting ADHD automatically. Our generated model is ready to be validated using a bigger database and it can also be used to detect other children's neurological disorders.

Sharper bounds for the error term in the prime number theorem

Sharper bounds for the error term in the prime number theorem

Remarks on Catalan's equation over function fields

Let ℓ be a prime number, F be a global function field of characteristic ℓ. Assume that there is a prime P∞ of degree 1. Let OF be the ring of functions in F with no poles outside of {P∞}. We study solutions to Catalan's equation Xm−Yn=1 over OF and show that under certain additional conditions, there are no non-constant solutions which lie in OF, when m,n>1.

Sign changes of coefficients of half-integral weight Hecke eigenforms

Let f \mathfrak {f} be a cusp form of half-weight k + 1 / 2 k+1/2 and at most quadratic nebentype character whose Fourier coefficients are denoted as a f ( n ) \mathfrak {a}_\mathfrak {f}(n) . We study sign changes of the family { a f ( t p 2 ) } p ∈ P \{\mathfrak {a}_\mathfrak {f}(tp^2)\}_{p\in \mathbb {P}} where t t is a square-free number and p p runs through the prime numbers. By taking use of the relationship between the cusp form f \mathfrak {f} of half-integral weight and the Shimura lift f t f_t , we establish that under certain conditions, the Hecke eigenforms of half-integral weight could be determined by sign changes of the sequence { a f ( t p 2 ) } p ∈ P \{\mathfrak {a}_\mathfrak {f}(tp^2)\}_{p\in \mathbb {P}} .

Galois-theoretic characterization of geometric isomorphism classes of quasi-monodromically full hyperbolic curves with small numerical invariants

Let l be a prime number. In the present paper, we prove that the geometric isomorphism class of a quasi-l-monodromically full hyperbolic curve with small numerical invariants over a sub-l-adic field is completely determined by the commensurability class of the kernel of the associated pro-l outer Galois action.

The average value of a certain number-theoretic function over the primes

We consider functions $F:\mathbb{Z}_{\geq 0}\rightarrow\mathbb{Z}_{\geq 0}$ for which there exists a positive integer $n$ such that two conditions hold: $F(p)$ divides $n$ for every prime $p$, and for each divisor $d$ of $n$ and every prime $p$, we have that $d$ divides $F(p)$ iff $d$ divides $F(p \mod d)$. Following an approach of Khrennikov and Nilsson, we employ the prime number theorem for arithmetic progressions to derive an expression for the average value of such an $F$ over all primes $p$, recovering a theorem of these authors as a special case. As an application, we compute the average number of $r$-periodic points of a multivariate power map defined on a product $Z_{f_1(p)}\times\cdots\times Z_{f_m(p)}$ of cyclic groups, where $f_i(t)$ is a polynomial.

A Review of Prime Numbers, Squaring Prime Pattern, Different Types of Primes and Prime Factorization Analysis

Abstract: The study of prime numbers and their properties has always been an intriguing and fascinating topic for mathematicians. Primes can be considered the “basic building blocks,” the atoms, of the natural numbers. They play a significant role in number theory. Also, prime numbers, in this current world of computers and digitalization, have paramount significance for the computer programmers and scientists to tackle relevant real-life problems. Since long time, many studies and researches have been conducted regarding prime numbers pattern. In this paper, a squaring prime pattern is presented. Moreover, fifteen different types of primes with their Python code to generate them is included within. In cryptographic encryption system, prime numbers play a major role for security systems in which prime factorization is necessary. Therefore, prime factorization of composite numbers using Sieve of Eratosthenes algorithm on different platforms and time analysis based on that has been presented in the paper. Also, factorization analysis of primes by five different algorithms has been shown and comparison of prime factorization of composite numbers vs time taken graph has been plotted. Two major applications of primes are also covered.

Indecomposable solutions of the Yang–Baxter equation of square-free cardinality

Indecomposable involutive non-degenerate set-theoretic solutions (X,r) of the Yang–Baxter equation of cardinality p1⋯pn, for different prime numbers p1,…,pn, are studied. It is proved that they are multipermutation solutions of level ≤n. In particular, there is no simple solution of a non-prime square-free cardinality. This solves a problem stated in [11] and provides a far reaching extension of several earlier results on indecomposability of solutions. The proofs are based on a detailed study of the brace structure on the permutation group G(X,r) associated to such a solution. It is proved that p1,…,pn are the only primes dividing the order of G(X,r). Moreover, the Sylow pi-subgroups of G(X,r) are elementary abelian pi-groups and if Pi denotes the Sylow pi-subgroup of the additive group of the left brace G(X,r), then there exists a permutation σ∈Symn such that Pσ(1),Pσ(1)Pσ(2),…, Pσ(1)Pσ(2)⋯Pσ(n) are ideals of the left brace G(X,r) and G(X,r)=P1P2⋯Pn. In addition, indecomposable solutions of cardinality p1⋯pn that are multipermutation of level n are constructed, for every nonnegative integer n.

Elements of Algorithmic Thinking in the Teaching of School Geometry through the Application of Geometric Problems

Algorithmic thinking and the creation of algorithms have traditionally been associated with mathematics. It is based on the general perception of an algorithm as a logically unambiguous and precise prescription for performing a certain set of operations, through which we reach a result in real time in a finite number of steps. There are well-known examples from history, such as the division algorithm used by ancient Babylonian mathematicians, Eratosthenes algorithm for finding prime numbers, Euclid’s algorithm for finding the greatest common divisor of two numbers, and cryptographic algorithm for coding and breaking, invented by Arabic mathematicians in the 9th century. Although the usage of algorithms and the development of algorithmic thinking currently fall within the domain of computer science, algorithms still play a role in mathematics and its teaching today. Contemporary mathematics, and especially its teaching in schools of all grades, prefers specific algorithms in arithmetic, algebra, and calculus. For example, operations with numbers, modifications of algebraic expressions, and derivation of functions. Teaching geometry in schools involves solving a variety of problems, many of which are presented as word problems. Algorithmization of school geometric tasks is therefore hardly visible and possible at first glance. However, there are ways to solve examples of a certain kind and to establish a characteristic and common algorithmic procedure for them. Algorithmic thinking in geometry and the application of algorithms in the teaching of thematic parts of school geometry are specific issue that we deal with in this study. We will focus on a detailed analysis of the possibilities of developing algorithmic thinking in school geometry and the algorithmization of geometric tasks.

A tangent inequality over primes

UDC 511 We introduce a new Diophantine inequality with prime numbers. Let 1 < c < 10 9 . We show that, for any fixed θ > 1 , every sufficiently large positive number N , and a small constant ε > 0 , the tangent inequality | p 1 c tan θ ( log p 1 ) + p 2 c tan θ ( log p 2 ) + p 3 c tan θ ( log p 3 ) - N | < ε has a solution in prime numbers p 1 , p 2 , and p 3 .

Optimal few-weight codes and their subfield codes

Linear codes with a few weights are an important class of codes in coding theory and have attracted a lot of attention. Recently, Wu et al., constructed few-weight quaternary linear codes and their subfield codes by using simplicial complexes. In this paper, we will extend the results from binary case to arbitrarily odd prime number [Formula: see text]. More specifically, we have constructed [Formula: see text] classes of distance optimal linear codes, all of them meet the Griesmer bound.

On continued fraction partial quotients of square roots of primes

We show that for each positive integer a there exist only finitely many prime numbers p such that a appears an odd number of times in the period of continued fraction of p or 2p. We also prove that if p is a prime number and D=p or 2p is such that the length of the period of continued fraction expansion of D is divisible by 4, then 1 appears as a partial quotient in the continued fraction of D. Furthermore, we give an upper bound for the period length of continued fraction expansion of D, where D is a positive non-square, and factorize some family of polynomials with integral coefficients connected with continued fractions of square roots of positive integers. These results answer several questions recently posed by Miska and Ulas [MU].

Arithmetic progressions in certain subsets of finite fields

In this note, we focus on how many arithmetic progressions we have in certain subsets of finite fields. For this purpose, we consider the sets Sp={t2:t∈Fp} and Cp={t3:t∈Fp}, and we use the results on Gauss and Kummer sums. We prove that for any integer k≥3 and for an odd prime number p, the number of k-term arithmetic progressions in Sp is given byp22k+R, where|R|≤(k−24−k−22k−1)⋅p32+ck⋅p and ck is a computable constant depending only on k. The proof also uses finite Fourier analysis and certain types of Weil estimates. Also, we obtain some formulas that give the exact number of arithmetic progressions of length ℓ in the set Sp when ℓ∈{3,4,5} and p is an odd prime number. For ℓ=4,5, our formulas are based on the number of points on certain elliptic curves, and the error term is best possible due to the Sato-Tate conjecture.

The Collatz problem in [formula omitted] and [formula omitted

We study the analogue of the Collatz map in the polynomial ring Fp[x], for any prime number p, and the corresponding dynamical system. We show that every f∈Fp[x] is eventually periodic in this system, in a quadratic number of iterations in deg(f), and describe explicitly all corresponding cycles. This extends a result of Hicks, Mullen, Yucas and Zavislak, who studied the case p=2. We also study the Collatz map in the formal power series ring Fp[[x]], observe that in Fp[[x]] all but countably many power series generate divergent trajectories via iterations of this map, and characterize those power series that are eventually periodic.