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José María Grau

Publications and source records attributed to José María Grau.

9 recordsLinked to original sources
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.

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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 $λ(n) \mid k(n-1)$.

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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.

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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.

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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.

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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.

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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.

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