Cullen numbers and Woodall numbers in generalized Fibonacci sequences (Q6556214)

For an integer \( k\ge 2 \), the \(k\)-generalized Fibonacci sequence, \( F_n^{(k)} \) is the linear recurrence sequence defined by\N\begin{align*}\NF_{n}^{(k)}=F_{n-1}^{(k)}+F_{n-1}^{(k)}+\cdots +F_{n-k}^{(k)},\N\end{align*}\Nwith the initial terms given by\N\begin{align*}\NF_{-(k-2)}^{(k)}=F_{-(k-2)}^{(k)}=\cdots =F_{0}^{(k)}=0\quad \text{and} \quad F_{1}^{(k)}=1.\N\end{align*}\NIn the paper under review, the authors study the Diophantine equations \N\[\NF_n^{(k)}=C_m,\tag{1}\N\]\Nand \N\[\NF_n^{(k)}=W_m,\tag{2}\N\]\Nin positive integers \( (m,n,k) \) with \( k\ge 2 \), where \( C_m=m\cdot 2^{m}+1 \) and \( W_m=m\cdot 2^{m}-1 \) are the Cullen numbers and Woodall numbers, respectively. Their main results are the following.\N\NTheorem 1. The Diophantine equation (1) has only the solutions \( F_1^{(k)}=F_2^{(k)}=1=C_0 \) for any \( k\ge 2 \), and the solution \( F_4^{(k)}=3=C_1 \). There are no other solutions in positive integers \( (m,n,k) \). \N\NTheorem 2. The Diophantine equation (2) has only the solutions \( F_1^{(k)}=F_2^{(k)}=1=W_1 \) for any \( k\ge 2 \), and the solutions \( F_{k+2}^{(k)}=2^{k}-1=W_m \) whenever \( m=2^{c} \) and \( k=c+2^{c} \) for some positive integer \( c \). There are no other solutions in positive integers \( (m,n,k) \).\N\NThe proofs of Theorem 1 and Theorem 2 follow from a clever combination of techniques in number theory, the usual properties of the \(k\)-generalized Fibonacci sequence and \( p \)-adic valuations. All calculations are done with the aid of computer program in \texttt{PARI/GP}.