Abstract:
In this study, we find all Pell and Pell–Lucas numbers that are product of two
repdigits in the base
$b$
for
$b\in[2,10]$.
It is shown that the largest Pell and
Pell–Lucas numbers that can be expressed as a product of two repdigits are
$P_{7}=169$
and
$Q_{6}=198$,
respectively.
Also, we have the representations
$$
P_{7}=169=(111)_{3}\times(111)_{3}$$
and
$$
Q_{6}=198=2\times99=3\times66=6\times33=9\times22.
$$
Furthermore, it is shown in the paper that the equation
$P_{k}=(b^{n}-1)(b^{m}-1)$
has only the
solution
$(b,k,m,n)=(2,1,1,1)$
and the equation
$Q_{k}=(b^{n}-1)(b^{m}-1)$
has no
solution
$(b,k,m,n)$
in positive integers for
$2\leq$$b\leq10$.
The proofs depend on
lower bounds for linear forms and some tools from Diophantine approximation.
Keywords:Pell number, Pell–Lucas number, repdigit, Diophantine equation, linear form in
logarithms.
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