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National and Regional Contests
Poland Contests
Polish MO Finals
1996 Polish MO Finals
2
Find all n such that x_n = 111111
Find all n such that x_n = 111111
Source:
October 30, 2005
induction
number theory unsolved
number theory
Problem Statement
Let
p
(
k
)
p(k)
p
(
k
)
be the smallest prime not dividing
k
k
k
. Put
q
(
k
)
=
1
q(k) = 1
q
(
k
)
=
1
if
p
(
k
)
=
2
p(k) = 2
p
(
k
)
=
2
, or the product of all primes
<
p
(
k
)
< p(k)
<
p
(
k
)
if
p
(
k
)
>
2
p(k) > 2
p
(
k
)
>
2
. Define the sequence
x
0
,
x
1
,
x
2
,
.
.
.
x_0, x_1, x_2, ...
x
0
,
x
1
,
x
2
,
...
by
x
0
=
1
x_0 = 1
x
0
=
1
,
x
n
+
1
=
x
n
p
(
x
n
)
q
(
x
n
)
x_{n+1} = \frac{x_np(x_n)}{q(x_n)}
x
n
+
1
=
q
(
x
n
)
x
n
p
(
x
n
)
. Find all
n
n
n
such that
x
n
=
111111
x_n = 111111
x
n
=
111111
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