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Baltic Way
2011 Baltic Way
16
16
Part of
2011 Baltic Way
Problems
(1)
Maximum exponent of prime dividing x_{2011}-1
Source: Baltic Way 2011
11/6/2011
Let
a
a
a
be any integer. Define the sequence
x
0
,
x
1
,
…
x_0,x_1,\ldots
x
0
,
x
1
,
…
by
x
0
=
a
x_0=a
x
0
=
a
,
x
1
=
3
x_1=3
x
1
=
3
, and for all
n
>
1
n>1
n
>
1
x
n
=
2
x
n
−
1
−
4
x
n
−
2
+
3.
x_n=2x_{n-1}-4x_{n-2}+3.
x
n
=
2
x
n
−
1
−
4
x
n
−
2
+
3.
Determine the largest integer
k
a
k_a
k
a
for which there exists a prime
p
p
p
such that
p
k
a
p^{k_a}
p
k
a
divides
x
2011
−
1
x_{2011}-1
x
2011
−
1
.
number theory proposed
number theory