Add to ORKGIn this article, we present four issues and provide a creative and concise proof for each of them. The four issues are: 1- Inequality $\frac{1}{\sqrt{n\pi+\frac{\pi}{2}}}<\frac{\binom{2n}{n}}{2^{2n}}<\frac{1}{\sqrt{n\pi}}$ 2- A special case of Jonathan Wilde's problem 3- Combination series 4- A feature of powerful numbers.Comment: 15 page
In this note we investigate the positive integersnfor whichφ(n2)+σ2(n) is divisible byn2
AbstractSome inequalities related to the Ky Fan and Wang inequalities for weighted arithmetic and ge...
AbstractRecently Andrews proposed a problem of finding a combinatorial proof of an identity on the q...
We deduce an inequality using elementary methods which makes it possible to prove a conjecture reg...
summary:We prove: If $A(n)$ and $G(n)$ denote the arithmetic and geometric means of the first $n$ po...
We give several integral representation for the Euler- Mascheroni constant using a combinatorial id...
AbstractA new expansion is given for partial sums of Eulerʼs pentagonal number series. As a corollar...
AbstractIn this paper, we study some inequalities involving the symmetric means. The main result is ...
Five binomial sums are extended by a free parameter $m$, that are shown, through the generating func...
AbstractNew lower bounds are given for the sum of degrees of simple and distinct irreducible factors...
AbstractBy using the Newton interpolation formula, we generalize the recent identities on the Catala...
We consider some finite binomial sums involving the derivatives of the binomial coefficient and deve...
We use an old elementary arithmetic argument to find new upper and lower bounds for Sylvester's denu...
Double inequalities for the remainder term of the Gauss quadrature formula are given. These inequa...
Let $A$ and $B$ be two subsets of the nonnegative integers. We call $A$ and $B$ additive complements...
In this note we investigate the positive integersnfor whichφ(n2)+σ2(n) is divisible byn2
AbstractSome inequalities related to the Ky Fan and Wang inequalities for weighted arithmetic and ge...
AbstractRecently Andrews proposed a problem of finding a combinatorial proof of an identity on the q...
We deduce an inequality using elementary methods which makes it possible to prove a conjecture reg...
summary:We prove: If $A(n)$ and $G(n)$ denote the arithmetic and geometric means of the first $n$ po...
We give several integral representation for the Euler- Mascheroni constant using a combinatorial id...
AbstractA new expansion is given for partial sums of Eulerʼs pentagonal number series. As a corollar...
AbstractIn this paper, we study some inequalities involving the symmetric means. The main result is ...
Five binomial sums are extended by a free parameter $m$, that are shown, through the generating func...
AbstractNew lower bounds are given for the sum of degrees of simple and distinct irreducible factors...
AbstractBy using the Newton interpolation formula, we generalize the recent identities on the Catala...
We consider some finite binomial sums involving the derivatives of the binomial coefficient and deve...
We use an old elementary arithmetic argument to find new upper and lower bounds for Sylvester's denu...
Double inequalities for the remainder term of the Gauss quadrature formula are given. These inequa...
Let $A$ and $B$ be two subsets of the nonnegative integers. We call $A$ and $B$ additive complements...
In this note we investigate the positive integersnfor whichφ(n2)+σ2(n) is divisible byn2
AbstractSome inequalities related to the Ky Fan and Wang inequalities for weighted arithmetic and ge...
AbstractRecently Andrews proposed a problem of finding a combinatorial proof of an identity on the q...