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1
Iran National Math Olympiad-Day 1-Problem 1
Iran National Math Olympiad-Day 1-Problem 1
Source:
April 30, 2010
algorithm
number theory
Euclidean algorithm
number theory unsolved
Problem Statement
Let
a
,
b
a,b
a
,
b
be two positive integers and
a
>
b
a>b
a
>
b
.We know that
gcd
(
a
−
b
,
a
b
+
1
)
=
1
\gcd(a-b,ab+1)=1
g
cd
(
a
−
b
,
ab
+
1
)
=
1
and
gcd
(
a
+
b
,
a
b
−
1
)
=
1
\gcd(a+b,ab-1)=1
g
cd
(
a
+
b
,
ab
−
1
)
=
1
. Prove that
(
a
−
b
)
2
+
(
a
b
+
1
)
2
(a-b)^2+(ab+1)^2
(
a
−
b
)
2
+
(
ab
+
1
)
2
is not a perfect square.
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