Divisibility Properties by Multisection

The/?-adic order, vp(r), of r is the exponent of the highest power of a prime/? which divides r. We characterize the/?-adic order vp(Fn) of the F„ sequence using multisection identities. The method of multisection is a helpful tool in discovering and proving divisibility properties. Here it leads to invariants of the modulo p Fibonacci generating function for p ^ 5. The proof relies on some simple results on the periodic structure of the series Fn. The periodic properties of the Fibonacci and Lucas numbers have been extensively studied (e.g., [13]). (For a general discussion of the modulo m periodicity of integer sequences, see [8].) The smallest positive index n such that Fn = 0 (mod/?) is called the rank of apparition (or rank of appearance, or Fibonacci entry-point) of prime/? and is denoted by n(p). The notion of rank of apparition n(m) can be extended to arbitrary modulus m>2. The order of/? in i^(p) will be denoted by e = e(p) = vp(Fn(py) > 1. Interested readers might consult [6] and [9] for a list of relevant references on the properties of vp(F„). The main focus of this paper is the multisection based derivation of some important divisibility properties of Fn (Theorem A) and Ln (Theorem D). A result similar to Theorem A was obtained by Halton [4]. This latter approach expresses the/?-adic order of generalized binomial coefficients in terms of the number of "carries." Theorem A can be generalized to include other linear recurrent sequences and a proof without using generating functions was given in Exercise 3.2.2.11 of [6], The latter approach is implicitly based on multisections. The generating functions of the Fibonacci and Lucas numbers are