J. Grau

activity

20102025

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

9 citations · 36 across the 31 of their papers we have counts for

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18 papers · 1 filter

math.NT2016

Fast computation of the number of solutions to $x_{1}^{2} + \dots + x_{k}^{2} \equiv λ (mod n)$

Jose Maria Grau, A. Oller-marcen

In this paper we study the multiplicative function $ρ_{k, λ} (n)$ that counts the number of incongruent solutions of the equation $x_{1}^{2} + \dots + x_{k}^{2} \equiv λ (mod n)$ . In particular…

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

Counting invertible sums of squares modulo $n$ and a new generalization of Euler totient function

Catalina Calderon, Jose Maria Grau, A. Oller-Marcen +1

In this paper we introduce and study a family $Φ_{k}$ of arithmetic functions generalizing Euler's totient function. These functions are given by the number of solutions to the equat…

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

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

Antonio M. Oller-Marcén, José María Grau

In this work we consider the congruence $\sum_{j = 1}^{n - 1} j^{k (n - 1)} \equiv - 1 (mod n)$ for each $k \in N$ , thus extending Giuga's ideas for $k = 1$ . In particular, it is pr…

Fast computation of the number of solutions to x12​+⋯+xk2​≡λ(modn)

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

Counting invertible sums of squares modulo n and a new generalization of Euler totient function

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