Divisibility properties of certain recurrent sequences
The paper contributes to the divisibility theory of sequences of the form \( {\mathbf x} \) whose terms \( x(0), x(1), \ldots \) are recursively produced integers. The main result is that if \( x(0) \) and \( m > 1 \) are integers and \( F \) is a polynomial having integer valued coefficients then the sequence produced by means of the recursion \( x(n) = x(n-1)^{m} + F(n) \) for \( n = 1, 2, \ldots \) is ultimately periodic modulo \( m \) in the sense that a positive integer \( t \) exists for which \( x(n+t) = x(n) \bmod m \) for all sufficiently large \( n \). Subsidiary results concern the case in which \( F(n) = 1 \) identically, \( x(0) = 2 \) and \( p \) is prime and not a factor of \( x(p) \). For example, if \( x(1)x(2) \ldots x(p-3) \) is divisible by \( p \) the sequence \( {\mathbf x} \) contains unboundedly many members divisible by \( p \); if the product is not divisible by \( p \), \( {\mathbf x} \) contains no member divisible by \( p \). The presentation is embedded in the theory of sequences of the form \( [x(n)] \) where \( {\mathbf x} \) is a real number sequence and the square brackets denote the integer part.
scientific article; zbMATH DE number 2187239
scientific article