The Period, Rank, and Order of the (a,b)-Fibonacci Sequence Mod m

The Fibonacci sequence F = 0, 1, 1, 2, 3, 5, 8, . . . has intrigued mathematicians for centuries, as it seems there is no end to its many surprising properties. Of particular interest to us are its properties when reduced under a modulus. It is well known, for example, that F (mod m) is periodic, that the zeros are equally spaced, and that each period of F (mod m) contains exactly 1, 2, or 4 zeros. We’ll denote the period of F (mod m) by π(m). Formulas are known for computing π(m) based on the prime factorization of m, but if p is prime, there is no formula for π(p). However, certain divisibility relations hold: π(p) | p − 1 if p ≡ ±1 (mod 10), and π(p) | 2(p + 1) if p ≡ ±3 (mod 10). This paper arose from the realization that many of the modulo m properties of the Fibonacci sequence are also properties of a much larger class of sequences. Further, matrix methods offer elementary proofs for the general case that are no more difficult than for the Fibonacci sequence itself. For integers a and b, we define the (a, b)-Fibonacci sequence F as the sequence with initial conditions F0 = 0, F1 = 1, that satisfies the general second-order linear recurrence relation Fn = aFn−1 + bFn−2. So, for example, the (1, 1)-Fibonacci sequence is the classic case F = 0, 1, 1, 2, 3, 5, . . . , and the (3,−2)-Fibonacci sequence begins 0, 1, 3, 7, 15, 31, . . . . In general, F = 0, 1, a, a2 + b, a3 + 2ab, . . . . In this article, we examine the behavior of the (a, b)-Fibonacci sequence under a modulus. When reducing the (a, b)-Fibonacci sequence modulo m, we’ll assume m is chosen so that gcd(b,m) = 1. That way, the sequence is uniquely determined backward as well as forward. For instance, we can compute F−1 ≡ b−1 (mod m). Modulo m, any pair of residues completely determines the sequence F , and there are finitely many pairs of residues, so F is periodic. We denote the period of F (mod m) by π(m). The rank of apparition, or simply rank, of F (mod m) is the least positive r such that Fr ≡ 0 (mod m), and we denote the rank of F (mod m) by α(m). If Fα(m)+1 ≡ s (mod m), observe that the terms of F starting with index α(m), namely 0, s, as, (a2 + b)s, . . . , are exactly the initial terms of F multiplied by a factor of s. Finally, we consider the order of F (mod m), denoted byω(m), and definedω(m) = π(m)/α(m). We shall see soon that ω(m) is always an integer, and that ω(m) = ordm(Fα(m)+1), the multiplicative order of Fα(m)+1 modulo m. Other authors have not named this function, but its close connection with the multiplicative order of Fα(m)+1 makes the name “order” seem reasonable. Lucas studied the (a, b)-Fibonacci sequence extensively and in 1878 established foundational results on the rank [9, section XXV]. He assigned 1 = a2 + 4b and deduced that if 1 is a quadratic residue (that is, a nonzero perfect square) mod p, then α(p) | p − 1. Also, if 1 is a quadratic nonresidue (a residue that is not a perfect square), then α(p) | p + 1. Finally, if p | 1, then α(p) = p. These results were all