Related papers: On amicable numbers
Euler starts with a hypergeometric series F(a, b, c, x), and differentiates it to get a functional relation. This relation is today known as Euler's identity. Then he integrates to get another and ends up with something like Legendre…
Partitions of the set of primes are introduced based on the Chebyshev polynomials at rationals. The prime densities of all such partitions are established. Euler's Criterion for $SL(2,\mathbb Q)$ is formulated, which is the bridge between…
A class of good integers has been introduced by P. Moree in $1997$ together with the characterization of good odd integers. Such integers have shown to have nice number theoretical properties and wide applications. In this paper, a complete…
We define the $m$th-order Eulerian numbers with a combinatorial interpretation. The recurrence relation of the $m$th-order Eulerian numbers, the row generating function and the row sums of the $m$th-order Eulerian triangle are presented. We…
Recently, Araci-Acikgoz-Sen derived some interesting identities on weighted q-Euler polynomials and higher-order q-Euler polynomials from the applications of umbral calculus (See [1]). In this paper, we develop the new method of q-umbral…
Understanding the notion of a model is not always easy in logic courses. Hence, tools such as Euler diagrams are frequently applied as informal illustrations of set-theoretical models. We formally investigate Euler diagrams as an…
While there are many identities involving the Euler and Bernoulli numbers, they are usually proved analytically or inductively. We prove two identities involving Euler and Bernoulli numbers with combinatorial reasoning via up-down…
Leonhard Euler, the most prolific mathematician in history, contributed to advance a wide spectrum of topics in celestial mechanics. At the Saint Petersburg Observatory, Euler observed sunspots and tracked the movements of the Moon.…
Everyone knows that the Euler characteristic of a combinatorial manifold is given by the alternating sum of its numbers of simplices. It is shown that there are other linear combinations of the numbers of simplices which are combinatorial…
We prove Euler's theorem of number theory developing an argument based on quandles. A quandle is an algebraic structure whose axioms mimic the three Reidemeister moves of knot theory.
Recently, we have endowed various categories of groups with topologies. The purpose of this paper is to introduce on these categories others topologies which are statistically more suitable to study well-known problems in groups theory. We…
This is a translation from the Latin original, "De valoribus integralium a termino variabilis x=0 usque ad x=infinity extensorum" (1781). This is E675 in the Enestrom index. Euler wants to find the location of the end point of a clothoid, a…
In his book "250 Problems in Elementary Number Theory", W.Sierpinski shows that the numbers 1+2^(2^n)+2^(2^n+1) are divisible by 21; for n=1,2,.... In this paper, we prove a similar but more general result.Consider the natural numbers of…
In many models in economics or business a dominantly self-interested homo economicus is assumed. Unfortunately (or fortunately), humans are in general not homines economici as e.g. the ultimatum game shows. This leads to the fact that all…
We consider the (barotropic) Euler system describing the motion of a compressible inviscid fluid driven by a stochastic forcing. Adapting the method of convex integration we show that the initial value problem is ill-posed in the class of…
We define a new class of numbers based on the first occurrence of certain patterns of zeros and ones in the expansion of irracional numbers in a given basis and call them Sagan numbers, since they were first mentioned, in a special case, by…
In this note we will discuss Euler's solution of the simple difference equation that he gave in his paper{\it ``De serierum determinatione seu nova methodus inveniendi terminos generales serierum"} \cite{E189} (E189:``On the determination…
A pair of sequences of natural numbers is called planar if there exists a simple, bipartite, planar graph for which the given sequences are the degree sequences of its parts. For a pair to be planar, the sums of the sequences have to be…
The paper introduces the notions of an elliptic pair, an elliptic cycle and an elliptic list over a square free positive integer d. These concepts are related to the notions of amicable pairs of primes and aliquot cycles that were…
The purpose of this paper is to introduce basic concepts that are fundamental in the examination of composite moduli, while avoiding the notoriously difficult problem of prime-factorization. We introduce a new class of numbers, called…