For any positive integer $k\ge 2$, let ${\left(P_n^{\left(k\right)}\right)}_{n\ge 2-k}$ be the $k$-generalized Pell sequence which starts with $0,\cdots ,0,1$ ($k$ terms) with the linear recurrence $$P_n^{\left(k\right)}=2P_{n-1}^{\left(k\right)}+P_{n-2}^{\left(k\right)}+\cdots +P_{n-k}^{\left(k\right)}\text{for}n\ge 2.$$ Let ${\left(N_n\right)}_{n\ge 0}$ be Narayana’s sequence given by $$N_0=N_1=N_2=1\text{and}{N}_{n+3}=N_{n+2}+N_n.$$ The purpose of this paper is to determine all $k$-Pell numbers which are sums of two Narayana’s numbers. More precisely, we study the Diophantine equation $$P_p^{\left(k\right)}=N_n+N_m$$ in nonnegative integers $k$, $p$, $n$ and $m$.

This paper is concerned with non-trivial solvability in $p$-adic integers of systems of additive forms. Assuming that the congruence equation $a{x}^k+b{y}^k+c{z}^k\equiv d\left(modp\right)$ has a solution with $xyz\not\equiv 0\left(modp\right)$ we have proved that any system of $R$ additive forms of degree $k$ with at least $2·3^{R-1}·k+1$ variables, has always non-trivial $p$-adic solutions, provided $p\nmid k$. The assumption of the solubility of the above congruence equation is guaranteed, for example, if $p\>k^4$.

In this paper, following L. Carlitz we consider some special equations of $n$ variables over the finite field of $q$ elements. We obtain explicit formulas for the number of solutions of these equations, under a certain restriction on $n$ and $q$.

We consider systems of equations of the form $x_i+x_j=x_k$ and $x_i·x_j=x_k$, which have finitely many integer solutions, proposed by A. Tyszka. For such a system we construct a slightly larger one with much more solutions than the given one.

On the assumption of a Riemann hypothesis for certain Hasse-Weil L-functions, it is shewn that a quaternary cubic form f(x) with rational integral coefficients and non-vanishing discriminant represents through integral vectors x almost all integers N having the (necessary) property that the equation f(x)=N is soluble in every p-adic field ℚₚ. The corresponding proposition for quinary forms is established unconditionally.