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We provide a proof of Wilson's Theorem and Wolstenholme's Theorem based on a direct approach by Lagrange requiring only basic properties of the primes and the Binomial theorem. The goal is to show how similar the two theorems are by…

Quantifier-elimination or model-completeness of the affine part of some classical first order theories are proved.

In this note we shall give a new proof to a quadrature formulae due to Newton.

I present a simple, elementary proof of Morley's theorem, highlighting the naturalness of this theorem.

I give a proof of the uniform boundedness theorem that is elementary (i.e. does not use any version of the Baire category theorem) and also extremely simple.

We prove the Complete nontrivial cycle-intersection theorem for systems of permutations.

We give here a new proof of a Tauberian Theorem of complex Laplace transform using the Theory of measure and theory of function with bounded variations. However we deduce the simple proof of Prime Number Theorem.

Fermat's statement is equivalent to say that if $x$, $y$, $z$, $n$ are integers and $n>2$, then $z^{n}\gtrless x^{n}+y^{n}$. This is proved with the aid of numbers $\lambda $'s, of the form $\lambda =z/\rho $, with $1<\rho<z$, named…

This paper gives a bijective proof of Andrews' refinement of the Alladi-Schur theorem. Moreover, it demonstrates that the bijective framework introduced here can be used to reproduce and provide a bijective account of Andrews' recursive…

We give an explicit formulae for obtaining the translation symmetries in the cartesian product $X^N$, where $N$ is some positive integer and $X$ is some finite set. Moreover, we obtain some fundamental results from elementary number theory.

We present a proof of Moessner's theorem by double induction, using only basic rules of arithmetic. No prerequisite knowledge is assumed. Familiarity with summation is advised.

In this note, we combine ideas of several previous proofs in order to obtain a quite short proof of Gr\"otzsch theorem.

The analogue of Goldie's Theorem for prime rings is proved for rings graded by abelian groups, eliminating unnecessary additional hypotheses used in earlier versions.

We proved that there are infinitely many cousin primes.

In this paper we give an additive representation of the factorial, which can be proven by a simple quick analytical argument. We also present some generalizations, which are linked, on the one hand to an arithmetical theorem proven by Euler…

In this note, we present a probabilistic proof of the well-known finite geometric series. The proof follows by taking the moments of the sum and the difference of two independent exponentially distributed random variables.

We give new proofs of some well-known results from Invariant Theorey using the Kempf-Ness theorem.

We suggest approximating the distribution of the sum of independent and identically distributed random variables with a Pareto-like tail by combining extreme value approximations for the largest summands with a normal approximation for the…

We give elementary proofs of two theorems concerning bounds on the maximum argument of the eigenvalues of a product of two unitary matrices --- one by Childs \emph{et al.} [J. Mod. Phys., \textbf{47}, 155 (2000)] and the other one by Chau…

We prove a stronger version of a termination theorem appeared in the paper "On existence of log minimal models II". We essentially just get rid of the redundant assumptions so the proof is almost the same as in there. However, we give a…