Euler's Function Research Articles

New recurrences for Euler''s partition function

In this paper, the author invokes some consequences of the bisectional pentagonal number theorem to derive two linear recurrence relations for Euler's partition function $p(n)$. As a corollary of these results, we obtain an efficient method to compute the parity of Euler's partition function $p(n)$ that requires only the parity of $p(k)$ with $k \leq n/4$.

A fast algorithm for enumerating abc-triples on the logarithmic function derived from Euler function

A fast algorithm for enumerating abc-triples on the logarithmic function derived from Euler function

Finite differences of Euler's zeta function

Finite differences of Euler's zeta function

Equivalent formulations for steady periodic water waves of fixed mean-depth with discontinuous vorticity

In this work we prove the equivalence between three different weak formulations of the steady periodic water wave problem where the vorticity is discontinuous. In particular, we prove that generalised versions of the standard Euler and stream function formulation of the governing equations are equivalent to a weak version of the recently introduced modified-height formulation. The weak solutions of these formulations are considered in Holder spaces.

Modeling crack discontinuities without element‐partitioning in the extended finite element method

SummaryIn this paper, we model crack discontinuities in two‐dimensional linear elastic continua using the extended finite element method without the need to partition an enriched element into a collection of triangles or quadrilaterals. For crack modeling in the extended finite element, the standard finite element approximation is enriched with a discontinuous function and the near‐tip crack functions. Each element that is fully cut by the crack is decomposed into two simple (convex or nonconvex) polygons, whereas the element that contains the crack tip is treated as a nonconvex polygon. On using Euler's homogeneous function theorem and Stokes's theorem to numerically integrate homogeneous functions on convex and nonconvex polygons, the exact contributions to the stiffness matrix from discontinuous enriched basis functions are computed. For contributions to the stiffness matrix from weakly singular integrals (because of enrichment with asymptotic crack‐tip functions), we only require a one‐dimensional quadrature rule along the edges of a polygon. Hence, neither element‐partitioning on either side of the crack discontinuity nor use of any cubature rule within an enriched element are needed. Structured finite element meshes consisting of rectangular elements, as well as unstructured triangular meshes, are used. We demonstrate the flexibility of the approach and its excellent accuracy in stress intensity factor computations for two‐dimensional crack problems. Copyright © 2016 John Wiley & Sons, Ltd.

Application of the Euler's gamma function to a problem related to F. Carlson's uniqueness theorem

In his work on F. Carlson's uniqueness theorem for entire functions of exponential type, Q. I. Rahman [5] was led to consider an infinite integral and needed to determine the rate at which the integrand had to go to zero for the integral to converge. He had an estimate for it which he was content with, although it was not the best that could be done. In the present paper we find a result about the behaviour of the integrand at infinity, which is essentially best possible. Stirling's formula for the Euler's Gamma function plays an important role in its proof.

A novel numerical technique to obtain an accurate solution to the Thomas-Fermi equation

In this paper, a new algorithm based on the fractional order of rational Euler functions (FRE) is introduced to study the Thomas-Fermi (TF) model which is a nonlinear singular ordinary differential equation on a semi-infinite interval. This problem, using the quasilinearization method (QLM), converts to the sequence of linear ordinary differential equations to obtain the solution. For the first time, the rational Euler (RE) and the FRE have been made based on Euler polynomials. In addition, the equation will be solved on a semi-infinite domain without truncating it to a finite domain by taking FRE as basic functions for the collocation method. This method reduces the solution of this problem to the solution of a system of algebraic equations. We demonstrated that the new proposed algorithm is efficient for obtaining the value of $ y'(0)$ , $ y(x)$ and $ y'(x)$ . Comparison with some numerical and analytical solutions shows that the present solution is highly accurate.

Repdigits as Euler functions of Lucas numbers

Abstract We prove some results about the structure of all Lucas numbers whose Euler function is a repdigit in base 10. For example, we show that if Ln is such a Lucas number, then n < 10111 is of the form p or p2, where p3 | 10p-1 -1.

On Euler's function

By giving some new treatments we can improve a classical result of Walfisz (1963) on the asymptotic formula of Euler's function.

Fast computation of the partition function

In this paper, the author provides a method to compute the values of Euler's partition function p(n) that requires only the values of p(k) with k⩽n/2. This method is combined with Ewell's recurrence relation for the partition function p(n) to obtain a simple and fast computation of the value of p(n).

A Mean Value Inequality for Euler's Beta Function

A Mean Value Inequality for Euler's Beta Function

Solving Nonlinear Fractional Integro-Differential Equations of Volterra Type Using Novel Mathematical Matrices

In this paper, the operational matrix of Euler functions for fractional derivative of order β in the Caputo sense is derived. Via this matrix, we develop an efficient collocation method for solving nonlinear fractional Volterra integro-differential equations. Illustrative examples are given to demonstrate the validity and applicability of the proposed method, and the comparisons are made with the existing results.

수학원리와 특성 진단을 기반으로 한 공개키 RSA 알고리즘의 현장 적용 프로세스

The RSA public key encryption algorithm, a few, key generation, factoring, the Euler function, key setup, a joint expression law, the application process are serial indexes. The foundation of such algorithms are mathematical principles. The first concept from mathematics principle is applied from how to obtain a minority. It is to obtain a product of two very large prime numbers, but readily tracking station the original two prime number, the product are used in a very hard principles. If a very large prime numbers p and q to obtain, then the product is the two easy station, a method for tracking the number of p and q from n synthesis and it is substantially impossible. The RSA encryption algorithm, the number of digits in order to implement the inverse calculation is difficult mathematical one-way function and uses the integer factorization problem of a large amount. Factoring the concept of the calculation of the mod is difficult to use in addition to the problem in the reverse direction. But the interests of the encryption algorithm implementation usually are focused on introducing the film the first time you use encryption algorithm but we have to know how to go through some process applied to the field work This study presents a field force applied encryption process scheme based on public key algorithms attribute diagnosis.

On the equation φ(Xm- 1) = Xn- 1

Here, we show that the title equation has only finitely many positive integer solutions (X, m, n), where φ is the Euler function.

Classifying orbits of the affine group over the integers

For each $n=1,2,\ldots ,$ let $\text{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^{n}$ be the affine group over the integers. For every point $x=(x_{1},\ldots ,x_{n})\in \mathbb{R}^{n}$ let $\text{orb}(x)=\{\unicode[STIX]{x1D6FE}(x)\in \mathbb{R}^{n}\mid \unicode[STIX]{x1D6FE}\in \text{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^{n}\}.$ Let $G_{x}$ be the subgroup of the additive group $\mathbb{R}$ generated by $x_{1},\ldots ,x_{n},1$. If $\text{rank}(G_{x})\neq n$ then $\text{orb}(x)=\{y\in \mathbb{R}^{n}\mid G_{y}=G_{x}\}$. Thus, $G_{x}$ is a complete classifier of $\text{orb}(x)$. By contrast, if $\text{rank}(G_{x})=n$, knowledge of $G_{x}$ alone is not sufficient in general to uniquely recover $\text{orb}(x)$; as a matter of fact, $G_{x}$ determines precisely $\max (1,\unicode[STIX]{x1D719}(d)/2)$ different orbits, where $d$ is the denominator of the smallest positive non-zero rational in $G_{x}$ and $\unicode[STIX]{x1D719}$ is the Euler function. To get a complete classification, rational polyhedral geometry provides an integer $1\leq c_{x}\leq \max (1,d/2)$ such that $\text{orb}(y)=\text{orb}(x)$ if and only if $(G_{x},c_{x})=(G_{y},c_{y})$. Applications are given to lattice-ordered abelian groups with strong unit and to AF $C^{\ast }$-algebras.

Growth of the Sudler product of sines at the golden rotation number

We study the growth at the golden rotation number ω=(5−1)/2 of the function sequence Pn(ω)=∏r=1n|2sin⁡πrω|. This sequence has been variously studied elsewhere as a skew product of sines, Birkhoff sum, q-Pochhammer symbol (on the unit circle), and restricted Euler function. In particular we study the Fibonacci decimation of the sequence Pn, namely the sub-sequence Qn=|∏r=1Fn2sin⁡πrω| for Fibonacci numbers Fn, and prove that this renormalisation subsequence converges to a constant. From this we show rigorously that the growth of Pn(ω) is bounded by power laws. This provides the theoretical basis to explain recent experimental results reported by Knill and Tangerman (2011) [10].

ON THE EQUATION 𝜙(5m- 1) = 5n- 1

Abstract. Here, we show that the title equation has no positive integersolutions (m,n), where φ is the Euler function. 1. IntroductionLet φ(m) be the Euler function of the positive integer m. Problem 10626from the American Mathematical Monthly [6] asks to find all positive integersolutions (m,n) of the Diophantine equation(1) φ(5 m −1) = 5 n −1.To our knowledge, no solution was ever received to this problem. Here, weprove the following result.Theorem 1. Equation (1) has no positive integer solution (m,n).Results in this spirit appear in [3], [4], [6], [7], [8], [9], [10], [11].2. The proof ofTheorem 1For the proof, we make explicit the arguments from [9] together with somespecific features which we deduce from the factorizations of 5 k − 1 for smallvalues of k. Write(2) 5 m −1 = 2 α p α 1 1 ···p α r r .Thus,(3) φ(5 m −1) = 2 α−1 p α 1 −11 (p 1 −1)···p α r −1r (p r −1).We achieve the proof of Theorem 1 as a sequence of lemmas. The first oneis known but we give a proof of it for the convenience of the reader.Lemma2. In equation (1), m and n are not coprime.

Simultaneous Padé approximants to the Euler, exponential and logarithmic functions

Simultaneous Padé approximants to the Euler, exponential and logarithmic functions

ON THE TORSOS FOR SOME GROUP SCHEMES OF PRIME-POWER ORDER

Abstract: By the classification theorem by F. Oort and J. Tate [6], any group scheme of prime order is isomorphic to a group scheme Ga,b under the suitable choice of a and b. We computed the torsors for some kinds of group schemes Ga,b in [8], which is a joint work with T. Sekiguchi, as in the following way: denote by p a prime number and by m = φ(p−1) the value of the Euler function φ. Suppose p is a prime ideal lying over p (which splits completely in Z[ζ]), where ζ is a primitive (p − 1)-st root of the unity. In case p is principal, the sequence 0 → μp,B → G m m,B p −→ Gm,B → 0 is exact, and the Galois descent of μp,B is isomorphic to Ga,b under the suitable choice of a and b, thus one can compute the torsors for this kinds of group schemes. The non-principal case is solved by Y. Koide [3] by using our method. The aim of this paper is to study some group schemes of order a power of a prime number. In section from 1 to 3, we would like to review the main result of the papers [6] by F. Oort and J. Tate, [4] by Y. Koide and T. Sekiguchi, and [8] by T. Sekiguchi and Y. Toda. In section 4, we give our main result, namely, the torsor for the Galois descent of μpn,B.

Arithmetic properties of the sum of the first $$n$$ n values of the Euler function

We give an upper bound on the counting function of the set of \(n\) such that the summatory function of the Euler function up to \(n\) is a square. Our estimate improves upon a previous estimate provided by Florian Luca and A. Sankaranarayanan.