Date: Tue, 23 Sep 2003 16:55:53 -0400 (EDT)
From: Dan Hoey <Hoey@aic.nrl.navy.mil>
To: John Conway <conway@math.princeton.edu>
Subject: Re: [math-fun] non-square products of squares?
cc: math-fun <math-fun@mailman.xmission.com>

John Conway writes:

>    Exactly what are these the probabilities of?  I was using
> the probability that a given triple  a,b,ab  should be of form
> square,square,nonsquare.  Are yours the conditional probability
> that  ab  should be a non-square GIVEN that  a,b  are squares?

>    [I mention that there is a third natural probability around,
> namely the probability that  a^2.b^2  should be a nonsquare.]

I see!  I first calculated the second probability, and saw that it
wasn't what you were describing.  Also, the denominators aren't as
pretty, because they reflect the somewhat flukey number of squares.

So I switched to the third, and since it agreed with your number (1/6)
for A4 (and since I overlooked the 1/36 you got for Q12) I thought
that was what you meant.  So the numbers I reported were the
probability that aabb is a nonsquare.

Looking at the first probability I see it can't exceed min(s^2,1-s)
where  s  is the density of squares--that's at most tau^-2 ~ .382, and
it's even less because not every product of squares can be a
nonsquare.  Up to order 255, the only group that exceeds 1/6 is
SmallGroup(48,50), or

    <a,b,c : a^3 = b^2 = c^2 = (a b)^3 = (a c)^3 = (b c)^2
                 = (a b a^-1 c)^2 = 1> .

It has a probability of 5/24 that a,b,ab are square,square,nonsquare.

Dan