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Mathematical analysis reveals explicit Dirichlet densities of primes dividing Lucas sequence ranks across all integer parameters, completing prior partial characterizations.
Let [Formula: see text] be a Lucas sequence, [Formula: see text] be a prime, and [Formula: see text] be the rank of appearance of [Formula: see text] in [Formula: see text], that is, the least positive integer [Formula: see text] such that [Formula: see text], if it exists. We derive closed-form formulas for the Dirichlet density of primes [Formula: see text] for which [Formula: see text], where [Formula: see text] is a fixed integer. Our results complete the work of Sanna ([Formula: see text]) by covering all [Formula: see text] and all [Formula: see text].
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Joaquim Cera Da Conceicao (2026) studied this question.
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