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MathLinks Contest 4th
7.3
7.3
Part of
MathLinks Contest 4th
Problems
(1)
0473 Fibonacci 4th edition Round 7 p3
Source:
5/7/2021
Let
{
f
n
}
n
≥
0
\{f_n\}_{n \ge 0}
{
f
n
}
n
≥
0
be the Fibonacci sequence, given by
f
0
=
f
1
=
1
f_0 = f_1 = 1
f
0
=
f
1
=
1
, and for all positive integers
n
n
n
the recurrence
f
n
+
1
=
f
n
+
f
n
−
1
f_{n+1} = f_n + f_{n-1}
f
n
+
1
=
f
n
+
f
n
−
1
. Let
a
n
=
f
n
+
1
f
n
a_n = f_{n+1}f_n
a
n
=
f
n
+
1
f
n
for any non-negative integer
n
n
n
, and let
P
n
(
X
)
=
X
n
+
a
n
−
1
X
n
−
1
+
.
.
.
+
a
1
X
+
a
0
.
P_n(X) = X^n + a_{n-1}X^{n-1} + ... + a_1X + a_0.
P
n
(
X
)
=
X
n
+
a
n
−
1
X
n
−
1
+
...
+
a
1
X
+
a
0
.
Prove that for all positive integers
n
≥
3
n \ge 3
n
≥
3
the polynomial
P
n
(
X
)
P_n(X)
P
n
(
X
)
is irreducible in
Z
[
X
]
Z[X]
Z
[
X
]
.
algebra
4th edition