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Contests
National and Regional Contests
Kyrgyzstan Contests
Kyrgyzstan National Olympiad
2012 Kyrgyzstan National Olympiad
5
5
Part of
2012 Kyrgyzstan National Olympiad
Problems
(1)
Prove that $a_k-22$ where $a_{k+2} = a_{k+1}a_k + 1$
Source: Kyrgyzstan 2012, Problem 5
5/2/2013
The sequence of natural numbers is defined as follows: for any
k
≥
1
k\geq 1
k
≥
1
,
a
k
+
2
=
a
k
+
1
⋅
a
k
+
1
a_{k+2}= a_{k+1}\cdot a_k+1
a
k
+
2
=
a
k
+
1
⋅
a
k
+
1
. Prove that for
k
≥
9
k\geq 9
k
≥
9
the number
a
k
−
22
a_k-22
a
k
−
22
is composite.
modular arithmetic
number theory unsolved
number theory