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ITAMO
2007 ITAMO
5
5
Part of
2007 ITAMO
Problems
(1)
The n-th term of a sequence is coprime with n
Source: Italian National Olympiad 2007, problem 5
5/13/2007
The sequence of integers
(
a
n
)
n
≥
1
(a_{n})_{n \ge 1}
(
a
n
)
n
≥
1
is defined by
a
1
=
2
a_{1}= 2
a
1
=
2
,
a
n
+
1
=
2
a
n
2
−
1
a_{n+1}= 2a_{n}^{2}-1
a
n
+
1
=
2
a
n
2
−
1
. Prove that for each positive integer n,
n
n
n
and
a
n
a_{n}
a
n
are coprime.
number theory proposed
number theory