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National and Regional Contests
China Contests
(China) National High School Mathematics League
2013 China Second Round Olympiad
2
Sequence with infinitely many squares
Sequence with infinitely many squares
Source: Chin Second Round (A) 2013 Q2
May 15, 2016
number theory
Problem Statement
Let
u
,
v
u,v
u
,
v
be positive integers. Define sequence
{
a
n
}
\{a_n\}
{
a
n
}
as follows:
a
1
=
u
+
v
a_1=u+v
a
1
=
u
+
v
, and for integers
m
≥
1
m\ge 1
m
≥
1
,
{
a
2
m
=
a
m
+
u
,
a
2
m
+
1
=
a
m
+
v
,
\begin{array}{lll} \begin{cases} a_{2m}=a_m+u, \\ a_{2m+1}=a_m+v, \end{cases} \end{array}
{
a
2
m
=
a
m
+
u
,
a
2
m
+
1
=
a
m
+
v
,
Let
S
m
=
a
1
+
a
2
+
…
+
a
m
(
m
=
1
,
2
,
…
)
S_m=a_1+a_2+\ldots +a_m(m=1,2,\ldots )
S
m
=
a
1
+
a
2
+
…
+
a
m
(
m
=
1
,
2
,
…
)
. Prove that there are infinitely many perfect squares in the sequence
{
S
n
}
\{S_n\}
{
S
n
}
.
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