We consider positional numeration system with negative base $-\beta$, as introduced by Ito and Sadahiro. In particular, we focus on arithmetical properties of such systems when $\beta$ is a quadratic Pisot number. We study a class of roots $\beta >1$ of polynomials $x^2-mx-n$, $m\ge n\ge 1$, and show that in this case the set $\mathrm{Fin}(-\beta )$ of finite $(-\beta )$-expansions is closed under addition, although it is not closed under subtraction. A particular example is $\beta =\tau =\frac{1}{2}(1+\sqrt{5})$, the golden ratio. For such $\beta$, we determine the exact bound on the number of fractional digits...

Prvočísla a otázky s nimi spojené představují často jedny z nejtěžších problémů matematiky a mnohé z nich zůstávají stále otevřené. V tomto článku se zabýváme otázkou, jak blízko ke zvolenému číslu již můžeme nalézt nějaké prvočíslo. Na základě známých tvrzení lze vyslovit hypotézu, že z každého přirozeného čísla lze již změnou nejvýše dvou číslic získat prvočíslo. Úvahy,  kterými rozvíjíme známé výsledky, jsou čistě aritmetické povahy. Vyslovená hypotéza, která je závislá na hypotéze z (Hanson,...

We establish new combinatorial transcendence criteria for continued fraction expansions. Let $\alpha =[0;a_1,a_2,...]$ be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients ${\left(a_{\ell }\right)}_{\ell \ge 1}$ of $\alpha$ is not ‘too simple’ (in a suitable sense) and cannot be generated by a finite automaton.

This paper studies the descriptional complexity of (i) sequences over a finite alphabet ; and (ii) subsets of $N$ (the natural numbers). If ${\left(s\left(i\right)\right)}_{i\ge 0}$ is a sequence over a finite alphabet $\Delta$, then we define the $k$-automaticity of $s,A_s^k\left(n\right)$, to be the smallest possible number of states in any deterministic finite automaton that, for all $i$ with $0\le i\le n$, takes $i$ expressed in base $k$ as input and computes $s\left(i\right)$. We give examples of sequences that have high automaticity in all bases $k$ ; for example, we show that the characteristic...

For each natural number $n$ we determine the average order $\alpha \left(n\right)$ of the elements in a cyclic group of order $n$. We show that more than half of the contribution to $\alpha \left(n\right)$ comes from the $\varphi \left(n\right)$ primitive elements of order $n$. It is therefore of interest to study also the function $\beta \left(n\right)=\alpha \left(n\right)/\varphi \left(n\right)$. We determine the mean behavior of $\alpha$, $\beta$, $1/\beta$, and also consider these functions in the multiplicative groups of finite fields.

Let Γ ⊂ ℚ * be a finitely generated subgroup and let p be a prime such that the reduction group Γₚ is a well defined subgroup of the multiplicative group ₚ*. We prove an asymptotic formula for the average of the number of primes p ≤ x for which [ₚ*:Γₚ] = m. The average is taken over all finitely generated subgroups $\Gamma =⟨a₁,...,a_r⟩\subset \mathbb{Q}*$, with $a_i\in \mathbb{Z}$ and $a_i\le T_i$, with a range of uniformity $T_i>exp\left(4{\left(logxloglogx\right)}^{1/2}\right)$ for every i = 1,...,r. We also prove an asymptotic formula for the mean square of the error terms in the asymptotic formula with a similar...

We study values of the Euler function φ(n) taken on binary palindromes of even length. In particular, if ${ℬ}_{2\ell }$ denotes the set of binary palindromes with precisely 2ℓ binary digits, we derive an asymptotic formula for the average value of the Euler function on ${ℬ}_{2\ell }$.