haggy

Doron, it is evident that there are two options here:

1) You are severely pedagogically challenged

2) You are incorrect

So which of the two is it?

As for your simple questions: What a load of garbled rubbish!

In fairy land, much like those "proofs" that you think are correct.

Blah => Blah => You're talking (or typing) tripe just like I am now

Blah, blah blah.

By the above argument, Doron, you are wrong. Can't you see that "Blah, blah blah." is a perfectly valid mathematical argument in your world?

Doron, we love your MyWay_XOR_TheHighway attitude

The moron is back

You obviously don't know much about Fermat's Last Theorem.

I'm not a mathematician

Gee, I think we all find that surprising

You ramble on about how Mathematics has to conform to your changing apples and how x != x.

If you denote "The exact state that an apple is in" by x (if that were quantifiable in some manner), this does not imply that x != x. Your changing apple (no matter what the time) does not mean this in the slightest. What it would rather mean is that your apple is in a new state x' and not its original state. The best you could say is that x != x'.

And who's Doron Shadmi anyway?

Someone whose reponses are quite similar to yours.

A typical conversation with Doron (*must spell correctly so it's not deleted*) goes something like this:

Doron: "I see things a particular way and in that context the whole of mathematics must be incorrect"

 

Respondent (typically Matt Grime): *A sound argument refuting what Doron had to say*

 

Doron: "You just don't get it(condescending tone)' date=' you've got to look at the way I do!"

.

.

.

More sound arguments

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.

.

Doron: "By my dodgy vague arguments all of Mathematics is wrong so you had better change your view people!"

[/quote']

*bangs head against wall*

No dear, the Language of Mathematics cannot be understood, if concepts like Infinity, Limit, Number are not understood first, before any technical definition.

Why are you so averse to having what you call technical definitions. The whole point of definitions is to ensure absolute clarity concerning the issue at hand. If one doesn't maintain absolute clarity there is room for confusion. An example of this would be the following:

Say we were to try to prove some theorem about triangles and it started...

Let P be the centre of the triangle.

Now, to the person initially writing it this might seem absolutely clear in meaning. But does this person mean the incentre or the circumcentre etc?

Doron, dealing with the concepts of Infinity, Number and Limit in a formal manner isn't a cop-out. Rather it is a means for all involved to know exactly what is being considered at a certain point. This is where much of your explanations don't have any formal definitions that would ensure absolute clarity.

In short' date=' for the past 100 years, pure mathematicians play with fancy notations without understanding the deep meaning of them.

 

Furthermore, they are doing their best in order to avoid any exploration of these fundamental concepts (like, Infinity, Number, Limit, Logic, etc…), and if someone tries to touch them, then immediately he can hear the same old broken record: ”It is not Mathematics”, “It is a philosophy”, “You do not understand Mathematics”, etc…[/quote']

There is also another broken record, that of your condescending: "You are not capable of understanding my included middle reasoning" or "You are confined by your 0_XOR_1 reasoning"

Maybe you could consider looking at some Coding Theory. Any electronic communication is reliant on Codes, so it is a field that has direct applications.

What is the scope of the essay? Are you expected to give some sort of an overview of a particular field and its applications?

This is very useful. Can you explain why 8765^4321 mod 9 = 8^4321 mod 9?

Although I know 8765 mod 9 = 8.

It's just the basic rules of modular arithmetic. Google "modular arithmetic".

Remember' date=' I'm using the "=" sign instead of the congruence sign. It's just quicker to type =.

Similarily, why 8^2160 mod 9 = 8?

I didn't say that. I said:

8^2 = 64 = 1 mod 9

Therefore,

(8^2)^2160 = (1)^2160 mod 9

Wow, Doron, your arms are so big. I envy you.

Do your arms get tired from all the hand-waving?

1 Inch = 2.54cm

193cm = 76 inches

Thus he's 6'4", same here.

Following this...

if mod 9' date=' this mean that the base (b-1) = 9 =>> b = 10

 

I don't see base 10 anywhere, or is it something else?

 

And can you explain what do you mean by base? [/quote']

 

We're working in the decimal system aka base 10.

 

E.g. 156 = 1*10^2 + 5*10^1 + 6 * 10^0

So if we look at S(156),

S(156) = 1+5+6 = 2 mod 9

Now

156 = 1*10^2 + 5*10^1 + 6 * 10^0 = 1*1^2 + 5*1^1 + 6 * 1^0 mod 9

= 1 + 5 + 6 mod 9

The reasoning here can be used to prove a more general theorem/lemma.

 

 

Concerning the number of digits of C:

The number of digits of 8765^4321 is approx Log10[8765^4321] ie 17036

The sum of those digits is at most 9*17036 = 153324 ....A

Now the sum of the digits of a 6 digit number is at most 6*9 = 54 ....B

 

This sum, B, along the same reasoning as before, has to be = 8 mod 9

i.e. B is in {8,17,26,35,44,53}

The sum of the digits of each of these is 8. Thus C = 8

Let S(x) = Sum of the digits of x

Thm: S(x) = x mod 9 (more generally Sb(x) = x mod (b-1) if we are working in base b)

S(8765^4321) = 8765^4321 mod 9

= 8^4321 mod 9

8^2 = 64 = 1 mod 9

Therefore

S(8765^4321) = (8^1)*(8^2)^2160 mod 9

= 8 * (1^2160) mod 9

= 8 mod 9

Thought this might help a bit.

C = 8. Merely calculated using Mathematica.

Haven't had a chance to check for more general cases yet.

(ab)^-1=a^-1b^-1 for every a' date='b are in G

[/quote']

That's not the case unless G is Abelian

 

c^-1 is unique in G so all you need is to show that (b^-1 * a^-1)ab = e = ab(b^-1 * a^-1), which is simple.

2004 = 167*3*4

These are relatively prime to one another.

2196^n – 25^n – 180^n + 13^n = (13*167+25)^n -25^n -(167+13)^n +13^n

Therefore

2196^n – 25^n – 180^n + 13^n = 25^n -25^n -13^n +13^n (mod 167)

2196^n – 25^n – 180^n + 13^n = 0 (mod 167)

Likewise

2196^n – 25^n – 180^n + 13^n = 0 (mod 3)

2196^n – 25^n – 180^n + 13^n = 0 (mod 4)

Therefore

2196^n – 25^n – 180^n + 13^n = 0 (mod 2004)

but with the advent of RSA and other encryptions using one way functions, its impossible.

Not impossible, just a whole lot harder.

As an aside: If your certificate for your https site isn't verified by someone like Verisign, you'll be susceptible to a meet in the middle attack.

Concerning careers: I'm doing Maths, not in it for the money, more for the enjoyment of my career.

I haven't looked at this closely but you need to consider repetition of vertices. A path is a walk that has no vertices repeated.

I think your working will change to something like n "choose" j/2 etc.

Cryptology is just applied number theory. You'll also need a lot of discrete maths. I suppose.

A bit of knowledge of Algebra(finite fields) is useful when considering AES/Rijndael. For that matter "A bit of knowledge of Algebra" is useful, period.