Re: FLTMA: A little group theory


Chip Eastham wrote:
The Dougster wrote:
Chip Eastham wrote:
The Dougster wrote:
The Dougster wrote:
The Dougster wrote:
Chip Eastham wrote:
The Dougster wrote:


http://www.mathpages.com/home/kmath264.htm

I think I see this more clearly today. If some power of w == -1 mod z
then, knowing w^0 = 1, we have { 1, -1 } <= <w> and so |<-1>| divides
|<w>|, where <= means "is a subgroup of".
I think it would be interesting to make a targeted search for
solutions with n = 3, the smallest possible prime, or even
better, to develop a proof that no such solutions exist.

Using what we have already shown, we need coprime x,y,z
such that:

(z/y) mod x has order 3
(z/x) mod y has order 3
(x/y) mod z has order 6 & (x/y)^3 == -1 mod z

where 0 < x < y < z < x+y and xyz even.

Thus phi(x) and phi(y) must be divisible by 3, and phi(z)
must be divisible by 6. Thus z = 7, 9, 13, 14, 19, 18,
etc. are candidates.

OK. I have done that in Mathcad, which doesn't share nicely in Usenet,
and am working in VB4 now, so I can do something up which can be
shared.

Is that xyz even or x+y+z even?

With x,y,z pairwise coprime, the notion that exactly one of x,y,z is
even is expressed equally well by xyz even or x+y+z even, but I
meant the former.

I wonder if I have access to a language in which operators can be
overloaded to make the source code look more like abstract algebra and
less like binary arithmetic. Hmm.

I like Prolog because one freely defines the relations/predicates that
make up the language (and they are readily "overloaded" for different
datatype, albeit at the expense of weak/runtime typing). However
you may want to look at some "functional" languages like Haskell
or ML. Lisp and descendants like Scheme are more "function"
friendly, but I don't know how important infix notation is for you.

Can anyone in sci.math suggest an appropriate limit for such a search?
Would 32-bit integers do all right? I think that's 16-bits of value
before the table a*b in Z/nZ* begins to overflow. There may be other
limits. Shall we say, to z<65535?

Yes, I think this would be plenty interesting, to run z out to 16 bit
unsigned limits, implemented in 32-bits for taking powers, etc.

I wonder if Prolog or any language allows syntax like the following for
programmers like me with some cognitive deficits to deal with:

* A function definition *
The Function phi of integer x returns integer y such that
y = 0
for z = 1 to z-1
if y mod x > 0 then y = y + 1
next z
return y

Just wondering how flexible syntax can be. In discourse I hate
pronouns. I abhor them. In programming I love them. In discourse, they
are omnipresent. In programming, absent. How about that.

Anyway, like this, from FLTMA Explorer 3, *untested*?

------------VB 4.0 Source-------------------
Private Sub Exp_3_Click()
For z = 5 To 65536
For y = 4 To z
For x = 3 To y
f _
gcd(x, y) = 1 And _
gcd(y, z) = 1 And _
gcd(x, z) = 1 And _
(x Mod 2) + (y Mod 2) + (z Mod 2) = 2 And _
(x < y) And (y < z) And (z < (x + y)) And _
( phi(x) * phi(y) * phi(z) ) mod 3 = 0 Then

Find (0)

If _
(zxyz + zyxz) > 1 And _
zxyz = 3 And _
xyzx = 3 And _
ozxy = 3 And _
ozyx = 3 Then
Text1.Text = Text1.Text & 3 & z & y & x & vbCrLf
End If

Next x
Next y
Next z


End Sub

--------------------------------------End VB 4.0
Source------------------------------

Doug Goncz
Replikon Research
Seven Corners, VA 22044-0384

.

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