It would seem that (1) is not as strong as (2). Note
that (2) would rule out an algorithm with
and
.
This
would be something that says ``this is a square'' most of the time,
occasionally correctly identifying a pseudo-square. However, the
paper (p. 293) shows that (1) implies (2).
Suppose we are given an algorithm. We estimate
with
high probability using the techniques in Section 7.4.
To take a specific example, we will assume we find
,
.
We want to test whether
is a square. Run the algorithm on
for 1000 randomly chosen
.
If
is a square, the algorithm will say
``this is a square''
times. If
is a pseudo-square, the
answer will be ``this is a square''
times.