For an integer $k\ge 2$, let ${\left(n\right)}_n$ be the $k-$generalized Pell sequence which starts with $0,...,0,1$ ($k$ terms) and each term afterwards is given by the linear recurrence $n=2n-1+n-2+\cdots +n-k$. In this paper, we find all $k$-generalized Pell numbers with only one distinct digit (the so-called repdigits). Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for repdigits in the usual Pell sequence ${\left(P_n^{\left(2\right)}\right)}_n$. ...

The seminal theorem of Cobham has given rise during the last 40 years to a lot of work about non-standard numeration systems and has been extended to many contexts. In this paper, as a result of fifteen years of improvements, we obtain a complete and general version for the so-called substitutive sequences. Let $\alpha$ and $\beta$ be two multiplicatively independent Perron numbers. Then a sequence $x\in A^{\mathbb{N}}$, where $A$ is a finite alphabet, is both $\alpha$-substitutive and $\beta$-substitutive if and only if $x$ is ultimately...

Let $\left\{U_n\right\}=\left\{U_n(P,Q)\right\}$ and $\left\{V_n\right\}=\left\{V_n(P,Q)\right\}$ be the Lucas sequences of the first and second kind respectively at the parameters $P\ge 1$ and $Q\in \{-1,1\}$. In this paper, we provide a technique for characterizing the solutions of the so-called Bartz-Marlewski equation $$x^2-3xy+y^2+x=0,$$ where $(x,y)=(U_i,U_j)$ or $(V_i,V_j)$ with $i$, $j\ge 1$. Then, the procedure of this technique is applied to completely resolve this equation with certain values of such parameters.

Consider a recurrence sequence ${\left(x_k\right)}_{k\in \mathbb{Z}}$ of integers satisfying $x_{k+n}=a_{n-1}{x}_{k+n-1}+...+a₁x_{k+1}+a₀x_k$, where $a₀,a₁,...,a_{n-1}\in \mathbb{Z}$ are fixed and a₀ ∈ -1,1. Assume that $x_k>0$ for all sufficiently large k. If there exists k₀∈ ℤ such that $x_{k₀}<0$ then for each negative integer -D there exist infinitely many rational primes q such that $q|x_k$ for some k ∈ ℕ and (-D/q) = -1.

A natural number $n$ is said to be a $(k,r)$-integer if $n=a^{k}b$, where $k>r>1$ and $b$ is not divisible by the $r$th power of any prime. We study the distribution of such $(k,r)$-integers in the Piatetski-Shapiro sequence $\left\{\lfloor n^c\rfloor \right\}$ with $c>1$. As a corollary, we also obtain similar results for semi-$r$-free integers.

Let $G$ be a finite group $G$, $K$ a field of characteristic $p\ge 17$ and let $U$ be the group of units in $KG$. We show that if the derived length of $U$ does not exceed $4$, then $G$ must be abelian.

Let $k\ge 2$ and let ${\left(P_n^{\left(k\right)}\right)}_{n\ge 2-k}$ be the $k$-generalized Pell sequence defined by $$P_n^{\left(k\right)}=2P_{n-1}^{\left(k\right)}+P_{n-2}^{\left(k\right)}+\cdots +P_{n-k}^{\left(k\right)}$$ for $n\ge 2$ with initial conditions $$P_{-(k-2)}^{\left(k\right)}=P_{-(k-3)}^{\left(k\right)}=\cdots =P_{-1}^{\left(k\right)}=P_0^{\left(k\right)}=0,P_1^{\left(k\right)}=1.$$ In this study, we handle the equation $P_n^{\left(k\right)}=y^m$ in positive integers $n$, $m$, $y$, $k$ such that $k,y\ge 2,$ and give an upper bound on $n.$ Also, we will show that the equation $P_n^{\left(k\right)}=y^m$ with $2\le y\le 1000$ has only one solution given by $P_7^{\left(2\right)}={13}^2.$

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given...