Related papers: Proving Touchard's Theorem From Euler's Form
The problem of constructing a perfect Euler cuboid is reduced to a single Diophantine equation of the degree 12.
We geometrically prove that in a d-dimensional cube with edges of length n, the number of particular d-dimensional tetrahedrons are given by Eulerian numbers. These tetrahedrons tassellate the cube, In this way the sum of the cubes are the…
In this paper, we study a composition of exponential generating functions. We obtain new properties of this composition, which allow to distinguish prime numbers from composite numbers. Using the result of paper we get the known properties…
We derive an Ehrhart function for symbols from the Euler-MacLaurin formula with remainder.
The problem of finding perfect Euler cuboids or proving their non-existence is an old unsolved problem in mathematics. The third cuboid conjecture is the last of the three propositions suggested as intermediate stages in proving the…
In this paper, we mainly show that Euler sums of generalized hyperharmonic numbers can be expressed in terms of linear combinations of the classical Euler sums.
We establish a Liouville type theorem for some conformally invariant fully nonlinear equations
In this work we state a Theorem on number theory and apply it to solve some ordinary and partial differential equations.
To determine Euler numbers modulo powers of two seems to be a difficult task. In this paper we achieve this and apply the explicit congruence to give a new proof of a classical result due to M. A. Stern.
In this paper some new ways of generalizing perfect numbers are investigated, numerical results are presented and some conjectures are established.
An Euler cuboid is a rectangular parallelepiped with integer edges and integer face diagonals. An Euler cuboid is called perfect if its space diagonal is also integer. Some Euler cuboids are already discovered. As for perfect cuboids, none…
We propose a new integral based on Taylor measures, study its properties extensively, and we illustrate that it includes many concepts from mathematics as special cases. In particular, the new integral emerges as a generalization of the…
Let $\sigma(x)$ be the sum of the divisors of $x$. If $N$ is odd and $\sigma(N) = 2N$, then the odd perfect number $N$ is said to be given in Eulerian form if $N = {q^k}{n^2}$ where $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and…
This is a new proof of the tauberian theorem of complex Laplace transformation for getting a simple proof of the prime numbers theorem $\pi(x) \sim \frac{x}{\log(x)}$ for a largest real $x$.
We give a new proof of Lucas' Theorem in elementary number theory.
In this article, we use the Touchard identity in order to obtain new integral representations for Catalan numbers. The main idea consists in combining the identity with a known integral representation and resorting to the binomial theorem.…
Gallagher's ergodic theorem is a result in metric number theory. It states that the approximation of real numbers by rational numbers obeys a striking 'all or nothing' behaviour. We discuss a formalisation of this result in the Lean theorem…
We present some further results on Liouville type theorems for some conformally invariant fully nonlinear equations.
We deduce the Riemann-Roch type formula expressing the microlocal Euler class of a perfect complex of D-modules in terms of the Chern character of the associated symbol complex and the Todd class of the manifold from the Riemann-Roch type…
The use of Cauchy's method to prove Euler's well-known formula is an object of many controversies. The purpose of this paper is to prove that Cauchy's method applies for convex polyhedra and not only for them, but also for surfaces such as…