A. Oller-Marcén

activity

20082025

most citedOn the diameter of the commuting graph of the full matrix ring over the real numbers

9 citations · 26 across the 23 of their papers we have counts for

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math.NT2016

Computing solutions to the congruence $1^{n} + 2^{n} + \dots + n^{n} \equiv p (mod n)$

Max Alekseyev, Jose Maria Grau, Amtonio Oller-Marcen

It is well-known that the congruence $\sum_{i = 1}^{n} i^{n} \equiv 1 (mod n)$ has exactly five solutions: ${1, 2, 6, 42, 1806}$ . In this work, we characterize the solutions to the…

math.NT2014

A von Staudt-type formula for $z \in Z_{n} [i] \sum z^{k}$

Pedro Fortuny Ayuso, Jose Maria Grau, Antonio Oller-Marcen

In this paper we study the sum of powers in the Gaussian integers $G_{k} (n) := \sum_{a, b \in [1, n]} (a + bi)^{k}$ . We give an explicit formula for $G_{k} (n) (mod n)$ in t…

math.NT2014★ 1 cited

Fermat test with gaussian base and Gaussian pseudoprimes

Jose M. Grau, Manuel Rodríguez, A. Oller-Marcen +1

The structure of the group $(Z / n Z)^{⋆}$ and Fermat's little theorem are the basis for some of best-known primality testing algorithms. Many related concepts ari…

math.NT2013★ 5 cited

On the congruence $1^{m} + 2^{m} + \dots + m^{m} \equiv n (mod m)$ with $n ∣ m$

José María Grau, Antonio M. Oller-Marcén, Jonathan Sondow

We show that if the congruence above holds and $n ∣ m$ , then the quotient $Q := m / n$ satisfies $\sum_{p ∣ Q} \frac{Q}{p} + 1 \equiv 0 (mod Q)$ , where $p$ is prime. The only known s…

math.NT2012

On arithmetic numbers

Antonio M. Oller-Marcén

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…

math.NT2011

On a variant of Giuga numbers

Jose Maria Grau, Florian Luca, Antonio M. Oller-Marcen

In this paper, we characterize the odd positive integers $n$ satisfying the congruence $\sum_{j = 1}^{n - 1} j^{\frac{n - 1}{2}} \equiv 0 (mod n)$ . We show that the set of such positive…

Computing solutions to the congruence 1n+2n+⋯+nn≡p(modn)

A von Staudt-type formula for z∈Zn​[i]∑​zk