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On the $1/3$ density of odd ranked primes in Lucas sequencesKeywords: Prime density , Lucas sequence , Legendre symbol , number field. Abstract: It seems that companion Lucas sequences often have a two-third density of prime divisors. Is it indeed the case? And if so, why? In this paper we present heuristics that predict this $2/3$ proportion before actually proving that $2/3$ is the correct prime density for almost all companion Lucas sequences.
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