Related papers: Disquitiones analyticae super evolutione potestati…
Euler states without proof statements about the form of prime divisors of numbers of the form aa+Nbb. See Ed Sandifer's How Euler Did It, ``Factors of Forms'', December 2005 at http://www.maa.org/news/howeulerdidit.html for a summary of the…
In this work, we illustrate and explore the use of Taylor series as solutions of differential equations. For a large a number of classes of differential equations in the literature, there are plenty of sources where the well known Taylor…
Using the Thue-Siegel method, we obtain effective improvements on Liouville's irrationality measure for certain one-parameter families of algebraic numbers, defined by equations of the type $(t-a)Q(t)+P(t)=0$. We apply these to some…
Motivated by the classical Eulerian number, descent and excedance numbers in the hyperoctahedral groups, an triangular array from staircase tableaux and so on, we study a triangular array $[\mathcal {T}_{n,k}]_{n,k\ge 0}$ satisfying the…
This is an English translation from the Latin original of Leonhard Euler's ``Solutio facilior problematis Diophantei circa triangulum, in quo rectae ex angulis latera opposita bisecantes rationaliter exprimantur''. In this paper, Euler…
We prove a result about reducibility behaviour of Thue polynomials over the rationals that was conjectured by M\"uller. More precisely, we show that, apart from few explicitly given exceptions, these polynomials have only finitely many…
An analogue of Taylor's formula, which arises by substituting the classical derivative by a divided difference operator of Askey-Wilson type, is developed here. We study the convergence of the associated Taylor series. Our results…
According to a theorem of Poincare, the solutions to differential equations are analytic functions of (and therefore have Taylor expansions in) the initial conditions and various parameters providing the right sides of the differential…
In this paper we introduce and study a family $\Phi_k$ of arithmetic functions generalizing Euler's totient function. These functions are given by the number of solutions to the equation $\gcd(x_1^2+\ldots +x_k^2, n)=1$ with $x_1,\ldots,x_k…
An extension of the theory of the Iterated Logarithmic Algebra gives the logarithmic analog of a Sheffer or Appell sequence of polynomials. This leads to several examples including Stirling's formula and a logarithmic version of the…
Euler wants to find rational numbers (integers) x and y such that x+y is a square and x^2+y^2 is a fourth power. He parametrizes these with two other variables that satisfy certain equations.
We present a unified algebraic framework utilizing the formal Bell transform to bridge the Dirichlet convolution of arithmetic functions with the combinatorial structure of infinite Euler-type products. By analyzing the logarithmic…
Translation from the Latin original, "Demonstratio gemina theorematis Neutoniani, quo traditur relatio inter coefficientes cuiusvis aequationis algebraicae et summas potestatum radicum eiusdem" (1747). E153 in the Enestrom index. In this…
In 1763, Euler published "Dilucidationes de resistentia fluidorum" (Explanations on the resistance of fluids), a memoir that challenges the fluid resistance theories proposed by Isaac Newton and d'Alembert. Euler's work explores the…
We use some basic properties of binomial and Stirling numbers to prove that the Euler characteristic is, essentially, the unique numerical topological invariant for compact polyhedra which can be expressed as a linear combination of the…
This note provides truncated formulae with explicit error terms to compute Euler products over primes in arithmetic progressions of rational fractions. It further provides such a formula for the product of terms of the shape $F(1/p, 1/p^s)$…
In 1776, L. Euler proposed three methods, called prima methodus, secunda methodus and tertia methodus, to calculate formulae for double zeta values. However strictly speaking, his last two methods are mathematically incomplete and require…
In this short note, we give two proofs of the infinitude of primes via valuation theory and give a new proof of the divergence of the sum of prime reciprocals by Roth's theorem and Euler-Legendre's theorem for arithmetic progressions.
In this paper, we study nonlinear differential equations arising from Eulerian polynomials and their applications. From our study of nonlinear differential equations, we derive some new and explicit identities involving Eulerian and…
In this paper we investigate some interesting of the (h,q)-extension of Euler numbers and polynomials. Finally, we will give some relations between these numbers anf polynomials