- I was asked offlist

> What the difference between the two acceptable mod operators

Is the following a fair summary:

> then?

The modulus relation is an equivalence relation. It is one way of

describing the cosets of the ideal generated by the 'base' B.

x == y (mod B) means

x-y \in ideal (B) or equivalently

coset x(B) = coset y(B).

To create modulus operators (more than one is possible, so it must be

defined before use) from this modulus relation, each coset of the

ideal must have one distinguished member chosen.

Briefly I'll limit myself to just the ring Z[i], as it has no tricky

features. (Note that everything probably falls to pieces in a

non-PID, but I'm sure any PID should work.)

Marcel's and my choice is to choose the member of the coset with the

smallest modulus. You could almost say that we use the Division

algorithm to chose our distinguished member.

(Note - when you start using a non-ED our choice does not correspond

to any Division Algorithm, as there is no such algorithm!)

Mike's choice, one used by Gauss, is to chose the member which has no

imaginary component. This parallels the Z version of mod where the

range too is entirely real.

My algorithm finds how far to move in order to get to the smallest

modulus, Marcel's technique goes straight there and should be used in

practice. His most recent explanation is the clearest I think.

Mike's not posted an algorithm, but David did posts a theta(p^2)

running time brute force algorithm (the first 2 lines of his first

bit of pari code). I'm fairly sure a constant-time algorithm should

be possible just by doing the algebra and using a technique like

Marcel's...

I would guess, if you were to want to do many computations with these

reduced values, that the choice of a single real value might

sometimes be easier to manage than the 2 smaller components of the

complex choice. Of this I can't be sure, as I've not coded anything

in this regard except the toy gp script yesterday.

?

Phil

=====

--

"One cannot delete the Web browser from KDE without

losing the ability to manage files on the user's own

hard disk." - Prof. Stuart E Madnick, MIT.

So called "expert" witness for Microsoft. 2002/05/02

__________________________________________________

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http://shopping.yahoo.com >to prove RH with Javascript?

I'm actually attempting to solve RH using Java.

Jon Perry

perry@...

http://www.users.globalnet.co.uk/~perry/maths

BrainBench MVP for HTML and JavaScript

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-----Original Message-----

From: djbroadhurst [mailto:d.broadhurst@...]

Sent: 11 May 2002 01:46

To: primenumbers@yahoogroups.com

Subject: [PrimeNumbers] Re: Back to Jon's original question about

complex mods.

> equivalence relation

all arguments are equivalent to Jon

> cosets

we cosset him enough already

> ideal

to prove RH with Javascript?

> ring

rarely a ring of truth in his claims

> PID

..dling

> smallest modulus

always mod (typo)

> Division algorithm

world expert!

> ED

never been to Athens

> imaginary

definitely

> range

epsilon

> real

..pain

Sorry Jon, but I had to get that

off my chest all at once.

Normally civility will be resumed

as soon as possible..

David

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