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Czech-Polish-Slovak Match
2007 Czech-Polish-Slovak Match
2
2
Part of
2007 Czech-Polish-Slovak Match
Problems
(1)
Fibonacci sequence=>∀m; ∃ k: m|(a_k^4-a_k-2)
Source: Czech-Polish-Slovak Match 2007-P2
9/14/2011
The Fibonacci sequence is defined by
a
1
=
a
2
=
1
a_1=a_2=1
a
1
=
a
2
=
1
and
a
k
+
2
=
a
k
+
1
+
a
k
a_{k+2}=a_{k+1}+a_k
a
k
+
2
=
a
k
+
1
+
a
k
for
k
∈
N
.
k\in\mathbb N.
k
∈
N
.
Prove that for any natural number
m
,
m,
m
,
there exists an index
k
k
k
such that
a
k
4
−
a
k
−
2
a_k^4-a_k-2
a
k
4
−
a
k
−
2
is divisible by
m
.
m.
m
.
number theory proposed
number theory