On generalized Mersenne numbers, their interpretations and matrix generators. (English) Zbl 1441.11027

The numbers of the form \(M_n=2^n-1\) are known as Mersenne numbers due to Marin Mersenne who studied these numbers in the 17th century. The primes of the form \(M_n\) are called Mersenne primes. And if \(M_n\) is prime then \(n\) must be prime. Also the \(n\)-th Mersenne number is the optimal solution of the Hanoi tower problem where \(n\) is the number of discs.
In the paper under review the authors introduce a generalization of Mersenne numbers. For a fixed positive integer \(k\ge 3\) the generalized Mersenne numbers are defined by the second order linear recurrence relation;
\[ M(k,n) = kM(k,n-1)-(k-1)M(k,n-2) \]
where \(M(k,0)=0\) and \(M(k,1)=1\). It can be seen from the definition that if \(k=3\) then \(M(3,n)=M_n\). First a Binet-type formula for the generalized Mersenne numbers is obtained by solving the linear recurrence. After that the authors give some important identities.
In section 2, the matrix generator for generalized Mersenne numbers is considered as;
\[\begin{bmatrix} M(k,n+2) & M(k,n+1) \\ M(k,n+1) & M(k,n)\end{bmatrix} = \begin{bmatrix} k & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} k & 1 \\ -(k-1) & 0\end{bmatrix}^n \]
Also some basic results are obtained.
In the same section the authors mention the lower Hessenberg matrix of size \(n\). They prove that the determinant of the \(n\)-th Hessenberg matrix is equal to \(M(k,n+1)\).
Combinatorial interpretations of the generalized Mersenne numbers are given in section 3.


11B37 Recurrences
15B36 Matrices of integers
Full Text: DOI


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