Number Theory

    

On the Largest Prime Factor of the K-Generalized Lucas Numbers

Authors: Herbert Batte, Florian Luca

Let $(L_n^{(k)})_{ngeq 2-k}$ be the sequence of $k$--generalized Lucas numbers for some fixed integer $kge 2$ whose first $k$ terms are $0,ldots,0,2,1$ and each term afterwards is the sum of the preceding $k$ terms. For an integer $m$, let $P(m)$ denote the largest prime factor of $m$, with $P(0)=P(pm 1)=1$. We show that if $n ge k + 1$, then $P (L_n^{(k)} ) > (1/86) log log n$. Furthermore, we determine all the $k$--generalized Lucas numbers $L_n^{(k)}$ whose largest prime factor is at most $ 7$.

Comments: 14 Pages.

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[v1] 2023-11-19 02:46:21

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