Several generalizations on Wolstenholme Theorem

This paper generalizes Wolstenholme theorem on two aspects. The first generalization is a parameterized form: let p > k + 2, k ≥ 1, ∀t ∈ ℤ, then ${{(pt + p - 1)!} \over {(pt)!}}\mathop \sum \limits_{m = 0}^{k - 1} {( - 1)^m}\mathop \sum \limits_{1 \le {i_l} < \cdots < {i_{k - m}} \le p - 1} {{{p^{k - (m + 1)}}} \over {\mathop \prod \limits_{l = 1}^{k - m} (pt + {i_l})}} \equiv 0{\left( {\bmod {p^{k + 1}}} \right)^.}$ The second generalization is on composite number module: Let 1overa be the x in congruent equation ax ≡ 1(mod m)(1 ≤ x < m), if m ≥ 5, then $$\matrix{ {\sum\limits_{\scriptstyle (k,m) = 1, \hfill \atop \scriptstyle 1 \le j \le m \hfill} {{{\left( {{1 \over k}} \right)}^2}} } \hfill & \equiv \hfill & {{m \over 6}[2m\varphi (m) + \prod\limits_{p|m} {(1 - p)]{{(\bmod m)}^{\;;}}} } \hfill \cr {\sum\limits_{\scriptstyle (k,m) = 1, \hfill \atop \scriptstyle 1 \le j \le m \hfill} {{{\left( {{1 \over k}} \right)}^3}} } \hfill & \equiv \hfill & {{{{m^2}} \over 4}[m\varphi (m) + \prod\limits_{p|m} {(1 - p)](\bmod m){\;^;}} } \hfill \cr {\sum\limits_{\scriptstyle (k,m) = 1, \hfill \atop \scriptstyle 1 \le j \le m \hfill} {{{\left( {{1 \over k}} \right)}^4}} } \hfill & \equiv \hfill & {{m \over {30}}[6{m^3}\varphi (m) + 10{m^2}\prod\limits_{p|m} {(1 - p) - \prod\limits_{p|m} {(1 - {p^3})](\bmod m){\;^;}} } } \hfill \cr {\sum\limits_{\scriptstyle (k,m) = 1, \hfill \atop \scriptstyle 1 \le j \le m \hfill} {{{\left( {{1 \over k}} \right)}^r}} } \hfill & \equiv \hfill & {{m^r}\sum\limits_{d|m} {\mu (d){{\left( {{m \over d}} \right)}^{ - r}}\sum\limits_{k = 1}^{{m \over d}} {{k^r}(\bmod m){\;^.}} } } \hfill \cr } $$ Where φ(x) is Euler function , μ(x) is Möbius function.

Distribution of Composite Numbers and Determination of Prime Numbers

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.

Distribution of Composite Numbers and Determination of Prime Numbers

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.

Distribution of Composite Numbers and Determination of Prime Numbers

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.

Distribution of Composite Numbers and Determination of Prime Numbers

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.

Discrete Pseudo-Fractional Fourier Transform and Its Fast Algorithm

In this article, we introduce a new discrete fractional transform for data sequences whose size is a composite number. The main kernels of the introduced transform are small-size discrete fractional Fourier transforms. Since the introduced transformation is not, in the generally known sense, a classical discrete fractional transform, we call it discrete pseudo-fractional Fourier transform. We also provide a generalization of this new transform, which depends on many fractional parameters. A fast algorithm for computing the introduced transform is developed and described.

Some Results of The Coprime Graph of a Generalized Quaternion Group Q_4n

The Coprime graph is a graph from a finite group that is defined based on the order of each element of the group. In this research, we determine the coprime graph of generalized quaternion group Q_(4n) and its properties. The method used is to study literature and analyze by finding patterns based on some examples. The first result of this research is the form of the coprime graph of a generalized quaternion group Q_(4n) when n = 2^k, n an odd prime number, n an odd composite number, and n an even composite number. The next result is that the total of a cycle contained in the coprime graph of a generalized quaternion group Q_(4n) and cycle multiplicity when is an odd prime number is p-1.Keywords: Coprime graph, generalized quaternion group, order, path AbstrakGraf koprima merupakan graf dari dari suatu grup hingga yang didefiniskan berdasarkan orde dari masing-masing elemen grup tersebut. Pada penelitian ini akan dibahas tentang bentuk graf koprima dari grup generalized quaternion Q_(4n). Metode yang digunakan dalam penelitian ini adalah studi literatur dan melakukan analisis berdasarkan pola yang ditemukan dalam beberapa contoh. Adapun hasil pertama dari penelitian adalah bentuk graf koprima dari grup generalized quaternion Q_(4n) untuk kasus n = 2^k, n bilangan prima ganjil ganjil, n bilangan komposit ganjil dan n bilangan komposit genap. Hasil selanjutnya adalah total sikel pada graf koprima dari grup generalized quaternion dan multiplisitas sikel ketika bilangan prima ganjil adalah p-1.Kata kunci: Graf koprima, grup generalized quternion, orde

Elementary Number Theory Problems. Part II

Summary In this paper problems 14, 15, 29, 30, 34, 78, 83, 97, and 116 from [6] are formalized, using the Mizar formalism [1], [2], [3]. Some properties related to the divisibility of prime numbers were proved. It has been shown that the equation of the form p 2 + 1 = q 2 + r 2, where p, q, r are prime numbers, has at least four solutions and it has been proved that at least five primes can be represented as the sum of two fourth powers of integers. We also proved that for at least one positive integer, the sum of the fourth powers of this number and its successor is a composite number. And finally, it has been shown that there are infinitely many odd numbers k greater than zero such that all numbers of the form 22 n + k (n = 1, 2, . . . ) are composite.

PRIME NUMBER LAW. DEPENDENCE OF PRIME NUMBERS ON THEIR ORDINAL NUMBERS AND GOLDBACH – EULER BINARY PROBLEM USING COMPUTER

The article considers the methods of defining and finding the distribution of composite numbers CN, prime numbers PN, twins of prime numbers Tw and twins of composite numbers TwCN that do not have divisors 2 and 3 in the set of natural numbers - ℕ based on a set of numbers like Θ = {6∙κ ± 1, κ ∈ ℕ}, which is a semigroup in relation to multiplication. There has been proposed a method of obtaining primes by using their ordinal numbers in the set of primes and vice versa, as well as a new algorithm for searching and distributing primes based on a closedness of the elements of the set Θ. It has been shown that a composite number can be presented in the form of products (6x ± 1) (6y ± 1), where x, y ℕ - are positive integer solutions of one of the 4 Diophantine equations: . It has been proved that if there is a parameter λ of prime twins, then none of Diophantine equations P (x, y, λ) = 0 has positive integer solutions. There has been found the new distribution law of prime numbers π(x) in the segment [1 ÷ N]. Any even number is comparable to one of the numbers i.e. . According to the above remainders m, even numbers are divided into 3 types, each type having its own way of representing sums of 2 elements of the set Θ. For any even number in a segment [1 ÷ ν], where ν = (ζ−m) / 6, , there is a parameter of an even number; it is proved that there is always a pair of numbers that are elements of the united sets of parameters of prime twins and parameters of transition numbers , i.e. numbers of the form with the same λ, if the form is a prime number, then the form is a composite number, and vice versa.