Let u = (u(n))(n >= 0) be a Lucas sequence, that is, a sequence of integers satisfying u(0) = 0, u(1) = 1, and u(n) = a(1)un(-1)+ a(2)u(n-2) for every integer n >= 2, where a(1) and a(2) are fixed nonzero integers. For each prime number p with p inverted iota 2a(2)D(u), where Du := a(1)(2) + 4a(2), let rho(u)(p) be the rank of appearance of p in u, that is, the smallest positive integer k such that p | u(k). It is well known that rho(u)(p) exists and that p = (D-u | p) (mod rho(u)(p)), where (D-u | p) is the Legendre symbol. Define the index of appearance of p in u as tau(u)( p) := ( p - (D-u | p) / rho(u)( p). For each positive integer t and for every x > 0, let P-u(t, x) be the set of prime numbers p such that p <= x, p inverted iota 2a(2)D(u), and iota(u)( p) = t. Under the Generalized Riemann Hypothesis, and under some mild assumptions on u, we prove that #P-u(t, x) = A F-u(t) G(u)(t) x/log x + O-u (x (log x)(2) + x log log(3x)phi(t)(log x)(2)), for all positive integers t and for all x > t(3), where A is the Artin constant, F-u(center dot) is a multiplicative function, and G(u)(center dot) is a periodic function (both these functions are effectively computable in terms of u). Furthermore, we provide some explicit examples and numerical data.

On the index of appearance of a Lucas sequence / Sanna, Carlo. - In: RAMANUJAN JOURNAL. - ISSN 1382-4090. - 63:4(2024), pp. 1179-1198. [10.1007/s11139-023-00811-4]

### On the index of appearance of a Lucas sequence

#### Abstract

Let u = (u(n))(n >= 0) be a Lucas sequence, that is, a sequence of integers satisfying u(0) = 0, u(1) = 1, and u(n) = a(1)un(-1)+ a(2)u(n-2) for every integer n >= 2, where a(1) and a(2) are fixed nonzero integers. For each prime number p with p inverted iota 2a(2)D(u), where Du := a(1)(2) + 4a(2), let rho(u)(p) be the rank of appearance of p in u, that is, the smallest positive integer k such that p | u(k). It is well known that rho(u)(p) exists and that p = (D-u | p) (mod rho(u)(p)), where (D-u | p) is the Legendre symbol. Define the index of appearance of p in u as tau(u)( p) := ( p - (D-u | p) / rho(u)( p). For each positive integer t and for every x > 0, let P-u(t, x) be the set of prime numbers p such that p <= x, p inverted iota 2a(2)D(u), and iota(u)( p) = t. Under the Generalized Riemann Hypothesis, and under some mild assumptions on u, we prove that #P-u(t, x) = A F-u(t) G(u)(t) x/log x + O-u (x (log x)(2) + x log log(3x)phi(t)(log x)(2)), for all positive integers t and for all x > t(3), where A is the Artin constant, F-u(center dot) is a multiplicative function, and G(u)(center dot) is a periodic function (both these functions are effectively computable in terms of u). Furthermore, we provide some explicit examples and numerical data.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11583/2984823`