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Problems
Contests
National and Regional Contests
PEN Problems
PEN A Problems
59
59
Part of
PEN A Problems
Problems
(1)
A 59
Source:
5/25/2007
Suppose that
n
n
n
has (at least) two essentially distinct representations as a sum of two squares. Specifically, let
n
=
s
2
+
t
2
=
u
2
+
v
2
n=s^{2}+t^{2}=u^{2}+v^{2}
n
=
s
2
+
t
2
=
u
2
+
v
2
, where
s
≥
t
≥
0
s \ge t \ge 0
s
≥
t
≥
0
,
u
≥
v
≥
0
u \ge v \ge 0
u
≥
v
≥
0
, and
s
>
u
s>u
s
>
u
. Show that
gcd
(
s
u
−
t
v
,
n
)
\gcd(su-tv, n)
g
cd
(
s
u
−
t
v
,
n
)
is a proper divisor of
n
n
n
.
Divisibility Theory