A field
F
F
of characteristic
≠
2
\neq 2
is said to have finite Pythagoras number if there exists an integer
m
≥
1
m\geq 1
such that each nonzero sum of squares in
F
F
can be written as a sum of
≤
m
\leq m
squares, in which case the Pythagoras number
p
(
F
)
p(F)
of
F
F
is defined to be the least such integer. As a consequence of Pfister’s results on the level of fields,
p
(
F
)
p(F)
of a nonformally real field
F
F
is always of the form
2
n
2^n
or
2
n
+
1
2^n+1
, and all integers of such type can be realized as Pythagoras numbers of nonformally real fields. Prestel showed that values of the form
2
n
2^n
,
2
n
+
1
2^n+1
, and
∞
\infty
can always be realized as Pythagoras numbers of formally real fields. We will show that in fact to every integer
n
≥
1
n\geq 1
there exists a formally real field
F
F
with
p
(
F
)
=
n
p(F)=n
. As a refinement, we will show that if
n
,
m
≥
2
n,m\geq 2
and
k
≥
1
k\geq 1
are integers such that
2
m
≥
2
k
≥
n
2m\geq 2^{k}\geq n
, then there exists a uniquely ordered field
F
F
with
p
(
F
)
=
n
p(F)=n
and
u
(
F
)
=
u
~
(
F
)
=
2
m
u(F)=\tilde {u}(F)=2m
(resp.
u
(
F
)
=
u
~
(
F
)
=
∞
u(F)=\tilde {u}(F)=\infty
), where
u
u
(resp.
u
~
\tilde {u}
) denotes the supremum of the dimensions of anisotropic forms over
F
F
which are torsion in the Witt ring of
F
F
(resp. which are indefinite with respect to each ordering on
F
F
).