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National and Regional Contests
Greece Contests
Greece National Olympiad
2012 Greece National Olympiad
1
1
Part of
2012 Greece National Olympiad
Problems
(1)
Positive integer couples
Source: Greek MO 2012 - P1
5/12/2013
Let positive integers
p
,
q
p,q
p
,
q
with
gcd
(
p
,
q
)
=
1
\gcd(p,q)=1
g
cd
(
p
,
q
)
=
1
such as
p
+
q
2
=
(
n
2
+
1
)
p
2
+
q
p+q^2=(n^2+1)p^2+q
p
+
q
2
=
(
n
2
+
1
)
p
2
+
q
. If the parameter
n
n
n
is a positive integer, find all possible couples
(
p
,
q
)
(p,q)
(
p
,
q
)
.
parameterization
number theory proposed
number theory