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National and Regional Contests
PEN Problems
PEN L Problems
9
9
Part of
PEN L Problems
Problems
(1)
L 9
Source:
5/25/2007
Let
{
u
n
}
n
≥
0
\{u_{n}\}_{n \ge 0}
{
u
n
}
n
≥
0
be a sequence of positive integers defined by
u
0
=
1
,
u
n
+
1
=
a
u
n
+
b
,
u_{0}= 1, \;u_{n+1}= au_{n}+b,
u
0
=
1
,
u
n
+
1
=
a
u
n
+
b
,
where
a
,
b
∈
N
a, b \in \mathbb{N}
a
,
b
∈
N
. Prove that for any choice of
a
a
a
and
b
b
b
, the sequence
{
u
n
}
n
≥
0
\{u_{n}\}_{n \ge 0}
{
u
n
}
n
≥
0
contains infinitely many composite numbers.
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