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Problems
Contests
National and Regional Contests
PEN Problems
PEN M Problems
25
25
Part of
PEN M Problems
Problems
(1)
M 25
Source:
5/25/2007
Let
{
a
n
}
n
≥
1
\{a_{n}\}_{n \ge 1}
{
a
n
}
n
≥
1
be a sequence of positive integers such that
0
<
a
n
+
1
−
a
n
≤
2001
for all
n
∈
N
.
0 < a_{n+1}-a_{n}\le 2001 \;\; \text{for all}\;\; n \in \mathbb{N}.
0
<
a
n
+
1
−
a
n
≤
2001
for all
n
∈
N
.
Show that there are infinitely many pairs
(
p
,
q
)
(p, q)
(
p
,
q
)
of positive integers such that
p
>
q
p>q
p
>
q
and
a
q
∣
a
p
a_{q}\; \vert \; a_{p}
a
q
∣
a
p
.
Recursive Sequences