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Antonio M. Oller-Marcén

Publications and source records attributed to Antonio M. Oller-Marcén.

15 recordsLinked to original sources
arXiv2019

Further generalizations of the parallelogram law

In recent work by Alessandro Fonda, a generalization of the parallelogram law in any dimension $N\geq 2$ was given by considering the ratio of the quadratic mean of the measures of the $N-1$-dimensional diagonals to the quadratic mean of the measures of the faces of a parallelotope. In this paper, we provide a further generalization considering not only $(N-1)$-dimensional diagonals and faces, but the $k$-dimensional ones for every $1\leq k\leq N-1$.

math.MG↗
arXiv2017

Zoel García de Galdeano y su presencia en sociedades matemáticas de su época

In this paper, we present the work of international mathematical associationism carried out by the mathematician from Navarra (Spain) Zoel García de Galdeano (1846-1924) during over 30 years. Garcia de Galdeano was a member of many of the most important mathematical associations of his time, thus being able to be in touch with foreign mathematicians of high level and to exchange ideas, works and collaborations. This work of associationism has been unnoticed in biographical and scientific studies up to now and helps to complete the picture of García de Galdeano as an international mathematician and as a wise importer of the main mathematical achievements of his time.

math.HO↗
arXiv2014

On the congruence $1^m + 2^m + \dotsb + m^m \equiv n \pmod{m}$ with $n | m$

We show that if the congruence above holds and $n\mid m$, then the quotient $Q:=m/n$ satisfies $\sum_{p\mid Q} \frac{Q}{p}+1 \equiv 0\pmod{Q}$, where $p$ is prime. The only known solutions of the latter congruence are $Q=1$ and the eight known primary pseudoperfect numbers $2,6,42, 1806, 47058, 2214502422, 52495396602,$ and $8490421583559688410706771261086$. Fixing $Q$, we prove that the set of positive integers $n$ satisfying the congruence in the title, with $m=Q n$, is empty in case $Q=52495396602$, and in the other eight cases has an asymptotic density between bounds in $(0,1)$ that we provide.

math.NT↗
arXiv2013

On the congruence $\sum_{j=1}^{n-1} j^{k(n-1)} \equiv -1 \pmod n $. k-strong Giuga and k-Carmichael numbers

In this work we consider the congruence $\sum_{j=1}^{n-1} j^{k(n-1)} \equiv -1 \pmod n$ for each $k \in \mathbb{N}$, thus extending Giuga's ideas for $k=1$. In particular, it is proved that a pair $(n,k)\in \mathbb{N}^2$ satisfies this congruence if and only if $n$ is prime or a Giuga Number and $λ(n) \mid k(n-1)$. In passing, we establish new characterizations of Giuga numbers and we study some properties of the numbers $n$ satisfying $λ(n) \mid k(n-1)$.

math.NT↗
arXiv2012

The $f$-belos

The \emph{arbelos} is the shape bounded by three mutually tangent semicircles with collinear diameters. Recently, Sondow introduced the parabolic analog, the \emph{parbelos} and proved several properties of the parbelos similar to properties of the arbelos. In this paper we give one step further and generalize the situation considering the figure bounded by (quite) arbitrary similar curves, the \emph{$f$-belos}. We prove analog properties to those of the arbelos and parbelos and, moreover, we characterize the parbelos and the arbelos as the $f$-beloses satisfying certain conditions.

math.HO↗
arXiv2012

On arithmetic numbers

An integer $n$ is said to be \textit{arithmetic} if the arithmetic mean of its divisors is an integer. In this paper, using properties of the factorization of values of cyclotomic polynomials, we characterize arithmetic numbers. As an application, in Section 2, we give an interesting characterization of Mersenne numbers.

math.NT↗
arXiv2012

Volumes of Solids of Revolution. A Unified Approach

We present a method to compute the volume of a solid of revolution as a double integral in a very simple way. Then, we see that the classical methods (disks and shells) are recovered if this double integral is computed by each of the two possible applications of Fubini's theorem. As a further application we also show how Pappus' theorem is obtained from our formula.

math.HO↗
arXiv2012

A determinant of generalized Fibonacci numbers

We evaluate a determinant of generalized Fibonacci numbers, thus providing a common generalization of several determinant evaluation results that have previously appeared in the literature, all of them extending Cassini's identity for Fibonacci numbers.

math.NT↗
arXiv2012

On $k$-Lehmer numbers

Lehmer's totient problem consists of determining the set of positive integers $n$ such that $φ(n)|n-1$ where $φ$ is Euler's totient function. In this paper we introduce the concept of $k$-Lehmer number. A $k$-Lehmer number is a composite number such that $φ(n)|(n-1)^k$. The relation between $k$-Lehmer numbers and Carmichael numbers leads to a new characterization of Carmichael numbers and to some conjectures related to the distribution of Carmichael numbers which are also $k$-Lehmer numbers.

math.NT↗
arXiv2012

On the last digit and the last non-zero digit of $n^n$ in base $b$

In this paper we study the sequences defined by the last and the last non-zero digits of $n^n$ in base $b$. For the sequence given by the last digits of $n^n$ in base $b$, we prove its periodicity using different techniques than those used by W. Sierpinski and R. Hampel. In the case of the sequence given by the last non-zero digits of $n^n$ in base $b$ (which had been studied only for $b=10$) we show the non-periodicity of the sequence when $b$ is an odd prime power and when it is even and square-free. We also show that if $b=2^{2^s}$ the sequence is periodic and conjecture that this is the only such case.

math.NT↗
arXiv2011

Tying up baric algebras

Given two baric algebras $(A_1,ω_1)$ and $(A_2,ω_2)$ we describe a way to define a new baric algebra structure over the vector space $A_1\oplus A_2$, which we shall denote $(A_1\bowtie A_2,ω_1\bowtieω_2)$. We present some easy properties of this construction and we show that in the commutative and unital case it preserves indecomposability. Algebras of the form $A_1\bowtie A_2$ in the associative, coutable-dimensional, zero-characteristic case are classified.

math.RA↗
arXiv2011

A primality test for $Kp^n+1$ numbers

In this paper we generalize the classical Proth's theorem for integers of the form $N=Kp^n+1$. For these families, we present a primality test whose computational complexity is $\widetilde{O}(\log^2(N))$ and, what is more important, that requires only one modular exponentiation similar to that of Fermat's test. Consequently, the presented test improves the most often used one, derived from Pocklington's theorem, which usually requires the computation of several modular exponentiations together with some GCD's.

math.NT↗
arXiv2011

Generalizing Giuga's conjecture

In 1950 G. Giuga studied the congruence $\sum_{j=1}^{n-1} j^{n-1} \equiv -1$ (mod $n$) and conjectured that it was only satisfied by prime numbers. In this work we generalize Giuga's ideas considering, for each $k \in \mathbb{N}$, the congruence $\sum_{j=1}^{n-1} j^{k(n-1)} \equiv -1$ (mod $n$). It particular, it is proved that a pair $(n,k)\in \mathbb{N}^2$ (with composite $n$) satisfies the congruence if and only if $n$ is a Giuga Number and $ λ(n)/\gcd(λ(n),n-1)$ divides $k$. In passing, we establish some new characterizations of Giuga Numbers.

math.NT↗
arXiv2011

Giuga Numbers and the arithmetic derivative

We characterize Giuga Numbers as solutions to the equation $n'=an+1$, with $a \in \mathbb{N}$ and $n'$ being the arithmetic derivative. Although this fact does not refute Lava's conjecture, it brings doubts about its veracity.

math.NT↗