[QUOTE=R.D. Silverman;217375]This is not what you wrote. You wrote:

"When you say so, you mean:
g1g2=b(mod m)
We get b."

I could not find a definition of g1 or g2 in your prior posts either.


For the last time, please stop inventing terminology (i.e. your expression
"agree the terms"). It is clear that "have their own inverse" is well
defined, so why invent terminology? It only adds confusion.



With regard to this last comment of yours: STOP! Don't say "agree the
terms" when you mean "unit equal to its own inverse".

Try proving the following:

(working mod m)
If x1 and x2 are units with order e1, prove that (x1*x2) has order e1.

Note that if x1 has order e1 and x2 has order e2, then the order of
(x1 * x2) is unpredictable.

So, let us return to the question: What determines when the product of
the units is 1 and what determines when it is -1.??

For this question you may restrict m to be squarefree. (otherwise things
can get complicated because of Hensel's Lemma [which we have not yet
discussed]).[/QUOTE]

Actually, let's skip this last question. It gets us too deep into group
theory and away from number theory. Instead, we simply observe,
for example, that integers m that have only 2 elements of order 2
must have the product of all the units equal to -1. Why? Because
the product of the units that don't have order 2 is 1 (we already
established this) and the units of order 2 are 1 and m-1, and their
product is -1.
It should also be easy to see that when the number of elements of
order 2 is a power of 2, then the product of all the units is +1.
(units of order 2 are closed under multiplication; pair them. then
pair their products. then pair those products etc.)

[quote=R.D. Silverman;217383]Actually, let's skip this last question. It gets us too deep into group
theory and away from number theory. Instead, we simply observe,
for example, that integers m that have only 2 elements of order 2
must have the product of all the units equal to -1. Why? Because
the product of the units that don't have order 2 is 1 (we already
established this) and the units of order 2 are 1 and m-1, and their
product is -1.
It should also be easy to see that when the number of elements of
order 2 is a power of 2, then the product of all the units is +1.
(units of order 2 are closed under multiplication; pair them. then
pair their products. then pair those products etc.)[/quote]
Let's discuss Hensel's lemma?

[QUOTE=blob100;217503]Let's discuss Hensel's lemma?[/QUOTE]

Premature.

[QUOTE=R.D. Silverman;217504]Premature.[/QUOTE]

BTW, you might actually want to try finishing the current problem.
We have shown that the product of the units whose order is not 2
equals 1. What remains is to show:

The product of units of order 2 is either 1 or -1.
Determine when this product is 1 and when it is -1.

This can actually be done fairly simply, without any group theory.

Hint: (-1)^n = 1 or -1 depending on n mod 2.

[quote=R.D. Silverman;217525]BTW, you might actually want to try finishing the current problem.
We have shown that the product of the units whose order is not 2
equals 1. What remains is to show:

The product of units of order 2 is either 1 or -1.
Determine when this product is 1 and when it is -1.

This can actually be done fairly simply, without any group theory.

Hint: (-1)^n = 1 or -1 depending on n mod 2.[/quote]

When you say "the product of units of order 2 is either 1 or -1." you mean that for units a1, a2, a3...,ak of order 2 we have (let's say) a1a2=+/-1(mod m)?
Or you mean: (a1a2)^2=+/-1(mod m)?

For example (I know you don't want me to give examples, but it will improve my understanding):
m=24.
a1=5 and a2=7 (we would say it can be 11 and 13 too, but let's take 5,7).
a1a2=35.
35-1=\=0(mod 24)
35+1=\=0(mod 24)
But 35^2=1(mod 24).

[QUOTE=blob100;217580]When you say "the product of units of order 2 is either 1 or -1." you mean that for units a1, a2, a3...,ak of order 2 we have (let's say) a1a2=+/-1(mod m)?
Or you mean: (a1a2)^2=+/-1(mod m)?
[/QUOTE]

For units a1,a2.....ak, of order 2, we have a1*a2*....*ak = 1 or -1

[/QUOTE]

We see that every product of two units ai of order 2 is antoher units of order 2.
The product of these three units (the two and the third we got by the product) is +/-1.
We may find K such "triangles" (set of three units discussed before).
There may be k such triangles -1.
The product of all units of order 2 will be denoted as g(m).
g(m)=(-1)^k(mod m).

[QUOTE=blob100;217629]We see that every product of two units ai of order 2 is antoher units of order 2.
The product of these three units (the two and the third we got by the product) is +/-1.
[/QUOTE]

This claim would need proof. And it is not true in general.
.[/QUOTE]

[quote=R.D. Silverman;217630]This claim would need proof. And it is not true in general.
.[/quote][/quote]
What do you mean by: "not true in general"?
You would just say: "False".
What is the direction?

[QUOTE=blob100;217634][/quote]
What do you mean by: "not true in general"?
You would just say: "False".
What is the direction?[/QUOTE]

Given three units of order 2, a1, a2, a3, it is [b]sometimes[/b]
true that a1*a2 *a3 = 1 or -1 mod m, but it is not always true.
Given any two such units a1, a2, we can find a unit a3, such
that a1 * a2 * a3 = 1 or -1, but it will not be true for (say)
a1 * a2 * a4 where a4 is a unit of order 2 different from a3.

It is true for [b]some[/b] triples of units, but not for all.

[quote=R.D. Silverman;217641]What do you mean by: "not true in general"?
You would just say: "False".
What is the direction?[/quote]

Given three units of order 2, a1, a2, a3, it is [B]sometimes[/B]
true that a1*a2 *a3 = 1 or -1 mod m, but it is not always true.
Given any two such units a1, a2, we can find a unit a3, such
that a1 * a2 * a3 = 1 or -1, but it will not be true for (say)
a1 * a2 * a4 where a4 is a unit of order 2 different from a3.

It is true for [B]some[/B] triples of units, but not for all.[/quote]

OK, This is the same as I said:
"We see that every product of two units ai of order 2 is antoher units of order 2.
The product of these three units (the two and the third we got by the product) is +/-1."