Abstract
AbstractLet $$(L_n^{(k)})_{n\ge 2-k}$$
(
L
n
(
k
)
)
n
≥
2
-
k
be the sequence of k–generalized Lucas numbers for some fixed integer $$k\ge 2$$
k
≥
2
whose first k terms are $$0,\ldots ,0,2,1$$
0
,
…
,
0
,
2
,
1
and each term afterward is the sum of the preceding k terms. For an integer m, let P(m) denote the largest prime factor of m, with $$P(0)=P(\pm 1)=1$$
P
(
0
)
=
P
(
±
1
)
=
1
. We show that if $$n \ge k + 1$$
n
≥
k
+
1
, then $$P (L_n^{(k)} ) > (1/86) \log \log n$$
P
(
L
n
(
k
)
)
>
(
1
/
86
)
log
log
n
. Furthermore, we determine all the k–generalized Lucas numbers $$L_n^{(k)}$$
L
n
(
k
)
whose largest prime factor is at most 7.
Funder
University of the Witwatersrand
Publisher
Springer Science and Business Media LLC
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