Let
K
K
be an algebraically closed field of characteristic different from
2
2
, let
g
g
be a positive integer, let
f
(
x
)
∈
K
[
x
]
f(x)\in K[x]
be a degree
2
g
+
1
2g+1
monic polynomial without multiple roots, let
C
f
:
y
2
=
f
(
x
)
\mathcal {C}_f: y^2=f(x)
be the corresponding genus
g
g
hyperelliptic curve over
K
K
, and let
J
J
be the Jacobian of
C
f
\mathcal {C}_f
. We identify
C
f
\mathcal {C}_f
with the image of its canonical embedding into
J
J
(the infinite point of
C
f
\mathcal {C}_f
goes to the zero of the group law on
J
J
). It is known [Izv. Math. 83 (2019), pp. 501–520] that if
g
≥
2
g\ge 2
, then
C
f
(
K
)
\mathcal {C}_f(K)
contains no points of orders lying between
3
3
and
2
g
2g
.
In this paper we study torsion points of order
2
g
+
1
2g+1
on
C
f
(
K
)
\mathcal {C}_f(K)
. Despite the striking difference between the cases of
g
=
1
g=1
and
g
≥
2
g\ge 2
, some of our results may be viewed as a generalization of well-known results about points of order
3
3
on elliptic curves. E.g., if
p
=
2
g
+
1
p=2g+1
is a prime that coincides with
char
(
K
)
\operatorname {char}(K)
, then every odd degree genus
g
g
hyperelliptic curve contains at most two points of order
p
p
. If
g
g
is odd and
f
(
x
)
f(x)
has real coefficients, then there are at most two real points of order
2
g
+
1
2g+1
on
C
f
\mathcal {C}_f
. If
f
(
x
)
f(x)
has rational coefficients and
g
≤
51
g\le 51
, then there are at most two rational points of order
2
g
+
1
2g+1
on
C
f
\mathcal {C}_f
. (However, there exist odd degree genus
52
52
hyperelliptic curves over
Q
\mathbb {Q}
that have at least four rational points of order 105.)