MathDB
Problems
Contests
International Contests
Baltic Way
2005 Baltic Way
17
17
Part of
2005 Baltic Way
Problems
(1)
Find all a which generates squares in the sequence
Source: Baltic Way 2005
12/28/2010
A sequence
(
x
n
)
n
≥
0
(x_n)_{n\ge 0}
(
x
n
)
n
≥
0
is defined as follows:
x
0
=
a
,
x
1
=
2
x_0=a,x_1=2
x
0
=
a
,
x
1
=
2
and
x
n
=
2
x
n
−
1
x
n
−
2
−
x
n
−
1
−
x
n
−
2
+
1
x_n=2x_{n-1}x_{n-2}-x_{n-1}-x_{n-2}+1
x
n
=
2
x
n
−
1
x
n
−
2
−
x
n
−
1
−
x
n
−
2
+
1
for all
n
>
1
n>1
n
>
1
. Find all integers
a
a
a
such that
2
x
3
n
−
1
2x_{3n}-1
2
x
3
n
−
1
is a perfect square for all
n
≥
1
n\ge 1
n
≥
1
.
number theory proposed
number theory