Antonio M. Oller-Marcén

On k-Lehmer Numbers

Abstract. Lehmers totient problem consists of determining the set of positive integers n such that where is Eulers totient function. In this paper we introduce the concept of k-Lehmer number. A k-Lehmer number is a composite number such that . 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.

On the congruence \(1^m + 2^m + \cdots + m^m\equiv n \pmod {m}\) with \(n\mid m\)

We show that if the congruence above holds and

ON POWER SUMS OF MATRICES OVER A FINITE COMMUTATIVE RING

n\mid m

Counting invertible sums of squares modulo

, then the quotient

n

Q:=m/n

and a new generalization of Euler's totient function

satisfies

The Best-or-Worst and the Postdoc problems

\sum_{p\mid Q} \frac{Q}{p}+1 \equiv 0\pmod{Q}

Power sums over finite commutative unital rings

, where

ON THE LAST DIGIT AND THE LAST NON-ZERO DIGIT OF n n IN BASE b

p

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

is prime. The only known solutions of the latter congruence are