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34
M 34
M 34
Source:
May 25, 2007
modular arithmetic
Recursive Sequences
Problem Statement
The sequence of integers
{
x
n
}
n
≥
1
\{ x_{n}\}_{n\ge1}
{
x
n
}
n
≥
1
is defined as follows:
x
1
=
1
,
x
n
+
1
=
1
+
x
1
2
+
⋯
+
x
n
2
(
n
=
1
,
2
,
3
⋯
)
.
x_{1}=1, \;\; x_{n+1}=1+{x_{1}}^{2}+\cdots+{x_{n}}^{2}\;(n=1,2,3 \cdots).
x
1
=
1
,
x
n
+
1
=
1
+
x
1
2
+
⋯
+
x
n
2
(
n
=
1
,
2
,
3
⋯
)
.
Prove that there are no squares of natural numbers in this sequence except
x
1
x_{1}
x
1
.
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