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Putnam
1987 Putnam
B3
B3
Part of
1987 Putnam
Problems
(1)
Putnam 1987 B3
Source:
8/5/2019
Let
F
F
F
be a field in which
1
+
1
≠
0
1+1 \neq 0
1
+
1
=
0
. Show that the set of solutions to the equation
x
2
+
y
2
=
1
x^2+y^2=1
x
2
+
y
2
=
1
with
x
x
x
and
y
y
y
in
F
F
F
is given by
(
x
,
y
)
=
(
1
,
0
)
(x,y)=(1,0)
(
x
,
y
)
=
(
1
,
0
)
and
(
x
,
y
)
=
(
r
2
−
1
r
2
+
1
,
2
r
r
2
+
1
)
(x,y) = \left( \frac{r^2-1}{r^2+1}, \frac{2r}{r^2+1} \right)
(
x
,
y
)
=
(
r
2
+
1
r
2
−
1
,
r
2
+
1
2
r
)
where
r
r
r
runs through the elements of
F
F
F
such that
r
2
≠
−
1
r^2\neq -1
r
2
=
−
1
.
Putnam