Silverman,
Will you give me other problems to solve please?

[QUOTE=blob100;215016]Silverman,
Will you give me other problems to solve please?[/QUOTE]

We have not yet finished solving this one. And I gave three more.
I have not seen any work on those.

[quote=R.D. Silverman;215026]We have not yet finished solving this one. And I gave three more.
I have not seen any work on those.[/quote]
2 is of order p modulo q.
For p,q deffined on the last posts:
2^p=1(mod q).
q|2^(2^n)+1.

I have a question:
In Shanks' book the concept of the cycle of a modulo m is posed.
Definition: If (a,m)=1 and a is of order e modulo m, the e residue classes a^1,a^2,a^3,...,a^e are called the cycle of a modulo m.
What is the meaning by saying: "the e residue classes a^1,a^2,a^3,...,a^e", the e residue classes of what?
Can this definition shown as a mathematical form?,
I mean:
If we have (a,m)=1
And a^e=1(mod m)
For a is of order e mod m,
We have:.....
Where we call "......" as the cycle of a modulo m.

Thanks.

[QUOTE=blob100;215417]I have a question:
In Shanks' book the concept of the cycle of a modulo m is posed.
Definition: If (a,m)=1 and a is of order e modulo m, the e residue classes a^1,a^2,a^3,...,a^e are called the cycle of a modulo m.
What is the meaning by saying: "the e residue classes a^1,a^2,a^3,...,a^e", the e residue classes of what?
[/QUOTE]
The integers a, a^2, a^3 .... a^e are each distinct. They form a set.
The size of this set is e. Each element of the set forms its own residue class mod m. This is perfectly clear.

I assume that you know what a residue class is.

[QUOTE]
Can this definition shown as a mathematical form?,
I mean:
If we have (a,m)=1
And a^e=1(mod m)
For a is of order e mod m,
We have:.....
Where we call "......" as the cycle of a modulo m.

Thanks.[/QUOTE]

These comments are gibberish.

[QUOTE=blob100;215040]2 is of order p modulo q.
For p,q deffined on the last posts:
2^p=1(mod q).
q|2^(2^n)+1.[/QUOTE]

I will wait two more days before posting the full answer.
Does anyone else want to weigh in?

Furthermore, Tomer, you have now shown [b]any[/b] work on the
other problems. Have you given up? Am I wasting my time?

[quote=R.D. Silverman;215442]I will wait two more days before posting the full answer.
Does anyone else want to weigh in?

Furthermore, Tomer, you have now shown [B]any[/B] work on the
other problems. Have you given up? Am I wasting my time?[/quote]
Yes.

Yawn.

Did "now" mean "not"?

David

[quote=R.D. Silverman;215442]I will wait two more days before posting the full answer.
Does anyone else want to weigh in?

Furthermore, Tomer, you have now shown [B]any[/B] work on the
other problems. Have you given up? Am I wasting my time?[/quote]

I will again tell you, I can't solve these problems.
I asked for others, maybe easiers, but you keep looking for the answers of these specified problems.

[QUOTE=blob100;215487]I will again tell you, I can't solve these problems.
I asked for others, maybe easiers, but you keep looking for the answers of these specified problems.[/QUOTE]

The one involving vectors is just high school trigonometry and geometry.
If you can't solve this one, then I am indeed wasting my time.

As for the other two: YOU HAVEN"T EVEN TRIED. I have seen no
ATTEMPT.

[quote=R.D. Silverman;215488]The one involving vectors is just high school trigonometry and geometry.
If you can't solve this one, then I am indeed wasting my time.

As for the other two: YOU HAVEN"T EVEN TRIED. I have seen no
ATTEMPT.[/quote]
I definietly can't understand problem number (3).
I have some questions About problem number (4):
How x is defined?
How a is defined?

[QUOTE=blob100;215490]I definietly can't understand problem number (3).
I have some questions About problem number (4):
How x is defined?
How a is defined?[/QUOTE]

For problem 3.

Consider a positive integer M. Do you know what a unit is?
Shanks' book discusses them. They are the integers that have
a multiplicative inverse mod M. This is a fundamental definition
and something you definitely [b]need[/b] to learn. phi(M)
calculates the number of units.

For example, for M = 6, the units are 1 and 5. For M=15, they are 1,2,4,7,8,11,13,14.

The problem asks: If one multiplies ALL of the units modulo M, then
what is the result?? Prove it.

Problem 4 simply asks: For a given integer M and a given integer a,
how many solutions are there to the equation: x^2 = a mod M.?
x is a VARIABLE. It will be, of course, an integer.