The quotient set of k-generalised Fibonacci numbers is dense in Q_p

The quotient set of A _ N is defined as R(A) := fa=b : a; b 2 A; b , 0g. Using algebraic number theory inQ(√p5), Garcia and Luca [‘Quotients of Fibonacci numbers’, Amer. Math. Monthly, to appear] proved that the quotient set of Fibonacci numbers is dense in the p-adic numbers Qp for all prime numbers p. For any integer k ≥2, let (F(k)n )n_≥(k-2) be the sequence of k-generalised Fibonacci numbers, defined by the initial values 0, 0,….0, 1 (k terms) and such that each successive term is the sum of the k preceding terms. We use p-adic analysis to generalise the result of Garcia and Luca, by proving that the quotient set of k-generalised Fibonacci numbers is dense in Qp for any integer k _ 2 and any prime number p.