Related papers: Euler sums and a prime numbers conundrum
The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order.…
Some class of sums which naturally include the sums of powers of integers is considered. A number of conjectures concerning a representation of these sums is made.
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 numbers of even and odd permutations with a given ascent number are investigated using an operator that was previously introduced by the author. Their difference is called a signed Eulerian number. By means of the operator the…
We announce a number of conjectures associated with and arising from a study of primes and irrationals in $\mathbb{R}$. All are supported by numerical verification to the extent possible.
In this paper we study the genralized q-Euler numbers and polynomials. From our results, we derive some interesting congruences related tothe generalized q-Euler numbers.
"Goldbach's Conjecture" proven by analysis of how all combinations of the odd primes, summed in pairs, generates all of the even numbers.
We give a complete and elementary proofs of "Jordan's sums" and study Euler's types sums. In particular we give a formula for the sum of series with same weight, which is similar to this one of classical 2-Euler's sums.
In this paper we introduce the generalization of Multi Poly-Euler polynomials and we investigate some relationship involving Multi Poly-Euler polynomials. Obtaining a closed formula for generalization of Multi Poly-Euler numbers therefore…
In this short note we present a class of conjectures on partitions of integers as summations of primes, which are extensions of Goldbach conjecture.
Translated from the Latin original "Facillima methodus plurimos numeros primos praemagnos inveniendi" (1778). E718 in the Enestrom index. If m is a number of the form 4k+1 and is a sum of two relatively prime squares, then it is prime if…
We study the distribution of primes from a topological viewpoint. Certain conjecture is introduced, and we show that it is equivalent to the Riemann Hypothesis.
In this paper, we deal with q-Euler numbers and q-Bernoulli numbers. We derive some interesting relations for q-Euler numbers and polynomials by using their generating function and derivative operator. Also, we show between the q-Euler…
The aim of this note is to provide a simple proof of some well-known identities and recurrences relating classical Bernoulli and Euler numbers by using the Abel sum of the divergent series $\sum_{n=0}^\infty (-1)^{n} (n+1)^k$, $k$ a…
We give explicit evaluations of the linear and non-linear Euler sums of hyperharmonic numbers $h_{n}^{\left( r\right) }$ with reciprocal binomial coefficients. These evaluations enable us to extend closed form formula of Euler sums of…
We prove explicit upper bounds for weighted sums over prime numbers in arithmetic progressions with slowly varying weight functions. The results generalize the well-known Brun-Titchmarsh inequality.
Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. The aim of this article is to give a result about the sum of euler's totient function from k equal 1 to n whene p divides n and p…
Legendre's Conjecture is one of the most elegant open problems in Number Theory, which states that there is a prime between consecutive two perfect squares. In this note, we prove the conjecture holds true and also discuss the related…
A survey paper on some recent results on additive problems with prime powers.
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…