# [Relationship between A002093 & A036913]

Joe Crump joecr at microsoft.com
Tue Feb 9 22:36:54 CET 1999

```Hi Christian & Sequence fans...,

I left something out, and should have been more careful
talking to a mathematical audience :).

Let me clarify...

The goal of f(n) is to find moduli which can deduce
information about composite numbers. The moduli 6 for instance,
yields 5 (xy mod 6 = 5), only when x & y are of the form 6i+1 & 6j+5.
However, there are no higher moduli than 6, that you can find
such a property where only 1 representation is possible.
For 2-representations, the highest m is 12.

In my original description of f(n) I wrote...
----------------------------------------------------------------------------
--------------------
f(n) = Maximum m where we can find (x,y), 0<x<m, 0<y<m, satisfying 0
< xy mod m <= n
----------------------------------------------------------------------------
--------------------

But for "xy mod m" I meant the count of particular solutions (xy mod
m) <= n.

An example is an order to clarify.

Where n = 1 and m = 6, we can construct the following unique
products {x,y} and
their result modulo 6...

-----------------------------------------------------
1*1                    1
1*2 2*2                2 4
1*3 2*3 3*3            3 0 3
1*4 2*4 3*4 4*4        4 2 0 4
1*5 2*5 3*5 4*5 5*5    5 4 3 2 1
-----------------------------------------------------

Now, the _counts_ of particular solutions (xy mod 6) are:

1 -> 2 (meaning occurs twice)
2 -> 3 (meaning occurs 3 times)
3 -> 3 (...)
4 -> 4
5 -> 1

Since 5 only occurs once, for 1*5, we know for all xy mod 6 = 5,
x & y are of the forms 6i+1 & 6j+5.

There are no moduli higher than 6 for which the count of
particular solutions (xy mod m) is 1. Sequentially, there are
no moduli higher than 12 for count(xy mod m) is 2... etc.

Anyone know a better way to write this than:
----------------------------------------------------------------------------
--------------------
f(n) = Maximum m where we can find (x,y), 0<x<m, 0<y<m, satisfying 0
< count(xy mod m) <= n
----------------------------------------------------------------------------
--------------------

Or better yet, the relationship with A002093 & A036913?

Cheers!

- Joe

-----Original Message-----
From: Christian G.Bower [mailto:bowerc at usa.net]
Sent: Sunday, February 07, 1999 5:24 PM
To: Joe Crump
Subject: Re: [Relationship between A002093 & A036913]

I think I am misunderstanding what you wrote:

> Here is the rule I followed to generate the sequence...
> f(n) = Maximum m where we can find (x,y), 0<x<m, 0<y<m,
> satisfying 0 < xy mod m <= n

Can't I choose x=y=1 so xy mod m = 1 <= n for all m,n>1?

Christian

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