Related papers: Proving Touchard's Theorem From Euler's Form

We call positive integer n a near-perfect number, if it is sum of all its proper divisors, except of one of them ("redundant divisor"). We prove an Euclid-like theorem for near-perfect numbers and obtain some other results for them.

We introduce Euler summability method for sequences of fuzzy numbers and state a Tauberian theorem concerning Euler summability method, of which proof provides an alternative to that of K. Knopp[\"Uber das Eulersche Summierungsverfahren II,…

In this article we give a result obtained of an experimental way for the Euler totient function.

While the general form of even perfect numbers is well-known, the existence or non-existence of odd perfect numbers is still an open problem. We address this problem and prove that if a natural number is odd, then it's not perfect.

We study the explicit formula of Euler numbers and polynomials of higher order

A perfect number is a number whose divisors add up to twice the number itself. The existence of odd perfect numbers is a millennia-old unsolved problem. This note proposes a proof of the nonexistence of odd perfect numbers. More generally,…

Translation from the Latin original, "Inventio summae cuiusque seriei ex dato termino generali" (1735). E47 in the Enestrom index. In this paper Euler derives the Euler-Maclaurin summation formula, by expressing y(x-1) with the Taylor…

Recently the new q-Euler numbers are defined. In this paper we derive the the Kummer type congruence related to q-Euler numbers and we introduce some interesting formulae related to these q-Euler numbers.

The Euler form is an Ext analog of the Euler characteristic, and in this paper we study the Euler form and give some applications. The first being a question of Jorgensen, which bounds the projective dimension of a module over a complete…

Using Eulerian and Euler numbers, we establish congruences concerning sums involving harmonic numbers, tangent numbers and Genocchi numbers.

Touchard-Riordan-like formulas are some expressions appearing in enumeration problems and as moments of orthogonal polynomials. We begin this article with a new combinatorial approach to prove these kind of formulas, related with integer…

We present a proof given by Euler in his paper {\it ``De serierum determinatione seu nova methodus inveniendi terminos generales serierum"} \cite{E189} (E189:``On the determination of series or a new method of finding the general terms of…

We introduce methods that allow to derive continuous-time versions of various discrete-time ergodic theorems. We then illustrate these methods by giving simple proofs and refinements of some known results as well as establishing new results…

We use Taylor's formula with Lagrange remainder to make a modern adaptation of Poisson's proof of a version of the fundamental theorem of calculus in the case when the integral is defined by Euler sums, that is Riemann sums with left (or…

A Tauberian theorem deduces an asymptotic for the partial sums of a sequence of non-negative real numbers from analytic properties of an associated Dirichlet series. Tauberian theorems appear in a tremendous variety of applications, ranging…

We establish a real version of Turrittin's result on polynomial and formal normal forms of linear systems of ODEs with meromorphic coefficients. Both the normal forms or the transformations used have only real coefficients. In order to…

In this paper, we study some properties of Euler polynomials arising from umbral calculus. Finally, we give some interesting identities of Euler polynomials using our results. Recently, Dere and Simsek have studied umbral calculus related…

We shall give some results for an integer divisible by its unitary totient.

The problem of finding perfect Euler cuboids or proving their non-existence is an old unsolved problem in mathematics. The second cuboid conjecture is one of the three propositions suggested as intermediate stages in proving the…

In this expository article we provide an elegant proof of the one-sided Ingham-Karamata Tauberian theorem. As an application, we present a short deduction of the prime number theorem.