Let $k\ge 2$ and define $F^{\left(k\right)}:={\left(F_n^{\left(k\right)}\right)}_{n\ge 0}$, the $k$-generalized Fibonacci sequence whose terms satisfy the recurrence relation $F_n^{\left(k\right)}=F_{n-1}^{\left(k\right)}+F_{n-2}^{\left(k\right)}+\cdots +F_{n-k}^{\left(k\right)}$, with initial conditions $0,0,\cdots ,0,1$ ($k$ terms) and such that the first nonzero term is $F_1^{\left(k\right)}=1$. The sequences $F:=F^{\left(2\right)}$ and $T:=F^{\left(3\right)}$ are the known Fibonacci and Tribonacci sequences, respectively. In 2005, Noe and Post made a conjecture related to the possible solutions of the Diophantine equation $F_n^{\left(k\right)}=F_m^{\left(\ell \right)}$. In this note, we use transcendental tools to provide a general method for finding the intersections $F^{\left(k\right)}\cap F^{\left(m\right)}$ which gives evidence supporting...

We completely solve the Diophantine equations $x²+2^a{q}^b=yⁿ$ (for q = 17, 29, 41). We also determine all $C=p{₁}^{a₁}\cdots p_k^{a_k}$ and $C=2^{a₀}p{₁}^{a₁}\cdots p_k^{a_k}$, where $p₁,...,p_k$ are fixed primes satisfying certain conditions. The corresponding Diophantine equations x² + C = yⁿ may be studied by the method used by Abu Muriefah et al. (2008) and Luca and Togbé (2009).

Let N be a set of natural numbers and Z be a set of integers. Let M₂(Z) denotes the set of all 2x2 matrices with integer entries. We give necessary and suficient conditions for solvability of the matrix negative Pell equation (P) X² - dY² = -I with d ∈ N for nonsingular X,Y belonging to M₂(Z) and his generalization (Pn) ${\sum }_{i=1}^{n}X{₂}_i-d{\sum }_{i=1}^{n}Y{²}_i=-I$ with d ∈ N for nonsingular $X_i,Y_i\in M₂\left(Z\right)$, i=1,...,n.