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 sequence of balancing numbers $\left(B_n\right)$ is defined by the recurrence relation $B_n=6B_{n-1}-B_{n-2}$ for $n\ge 2$ with initial conditions $B_0=0$ and $B_1=1.$$B_n$ is called the $n$th balancing number. In this paper, we find all repdigits in the base $b,$ which are sums of four balancing numbers. As a result of our theorem,...

We analyse the roots of the polynomial $x^n-p{x}^{n-1}-qx-1$ for $p⩾q⩾1$. This is the characteristic polynomial of the recurrence relation $F_{k,p,q}\left(n\right)=p{F}_{k,p,q}(n-1)+q{F}_{k,p,q}(n-k+1)+F_{k,p,q}(n-k)$ for $n⩾k$, which includes the relations of several particular sequences recently defined. In the end, a matricial representation for such a recurrence relation is provided.

In this paper we give a much shorter proof for a result of B.M.M de Weger. For this purpose we use the theory of linear forms in complex and $p$-adic elliptic logarithms. To obtain an upper bound for these linear forms we compare the results of Hajdu and Herendi and Rémond and Urfels.

We consider Diophantine quintuples a, b, c, d, e. These are sets of positive integers, the product of any two elements of which is one less than a perfect square. It is conjectured that there are no Diophantine quintuples; we improve on current estimates to show that there are at most $5.441·{10}^{26}$ Diophantine quintuples.

The goal of this article is twofold. First, we extend a result of Murty and Saradha (2007) related to the digamma function at rational arguments. Further, we extend another result of the same authors (2008) about the nature of p-adic Euler-Lehmer constants.