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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,676

Bob was having fun with his, why this Zeckendorf chap is not even a member!

Here is another six:

Doron Zeilberger;

Herbert S. Wilf

Abraham De Moivre of course

Newton of course

Freeman Dyson ( little known )

Pierre De Fermat of course

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,525

Here is a list of some the most known mathematicians:

Euclid

Archimedes

Leonhard Euler

John von Neumann

Pythagoras of Samos

Isaac Newton

Gottfried Wilhelm Leibniz

Rene Descartes

Pierre de Fermat

Carl Friedrich Gauss

Andrey Markov

Evangelista Torricelli

Leonardo Pisano Bigollo (better known as Fibonacci)

Edouard Lucas

Blaise Pascal

Stefan Banach

Srinisava Ramanujan

Brahmagupta (thank you for nothing )

Evariste Galois

William Rowan Hamilton

Donald Knuth

Doron Zeilberger

Herbert Wilf

Henri Poincare

David Hilbert

Joseph-Louis Lagrange

Bernhard Riemann

Niels Abel

Arthur Cayley

Augustin Cauchy

Georg Cantor

Peter Dirichlet

Diophantus of Alexandria

Pierre-Simon Laplace

Jean le Rond D'Alembert

Francois Viete

Joseph Fourier

Jacob Bernoulli

Simeon-Denis Poisson

There are a lot more great mathematicians, but the ones above should give you a pretty good choice.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,676

That is a few more than six.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,525

He can still pick the ones he would like. My only mistake is that I haven't included any females in the list. Candidates would be Sophie Germain and S. Bundy.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,676

Hypatia

Ada Lovelace

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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anonimnystefy wrote:

I tried reading your proof, but my tiny mind couldn't comprehend it.

Me too! May be I will understand some other day when I shall know Mathematics better

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'Humanity is still kept intact. It remains within.' -Alokananda

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,525

I just get lost in it. I will have to sit down one day and just read it carefully and thoroughly.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,676

Calm down, both of you will get there soon enough!

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,393

I'm supposed to be a teacher. Eeeekkk!

I'll have another go if you ask me to.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,525

You don't have to trouble yourself, if you ask me. Maybe I am just not careful reading your proof. I will have another go at it later today.

Here lies the reader who will never open this book. He is forever dead.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,393

hi Stefy,

That wasn't the answer I was hoping for. Perhaps I should have said "I will enjoy trying to make a better explanation".

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,525

Why would you type up a clearer explanation if the one we have now can be understood with a little bit more effort?

Here lies the reader who will never open this book. He is forever dead.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,393

Because you said you didn't understand it => it wasn't clear.

B

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,525

Incorrect!

I say that it is unclear && I made a reasonable ammount of effort => It is unclear.

And because both conditions aren't met, we cannot claim anything about the clarity of your proof.

Here lies the reader who will never open this book. He is forever dead.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,393

Actually you said your tiny mind couldn't comprehend it.

Since (as a teacher) I have the patience of a saint, I was prepared to spell it out in more simple terms, specifically taking account of your tiny mind. But if you are rejecting my offer, I'll just seek out someone who appreciates my talent.

Bob

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,525

Whoa, it seems to me you are twisting my words. I never said that I don't appreciate your talent! If it makes you happy, then, please, present your proof in simpler terms to me.

Here lies the reader who will never open this book. He is forever dead.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,393

Oh thank you! I'd love to.

The algorithm shows how you can reduce any counting number to a sum of Fibonacci numbers (without repetition).

A number N has an F number subtracted which is saved in a list.

Then the remainder (ie from the subtraction, no division involved) is treated as the new N and the algorithm repeated.

This continues until the remainder is an F; at which point the process stops and the list of subtracted Fs gives the required sum.

There are three things to prove:

(i) Will it always be possible to find an F to subtract?

(ii) How do I know that no F is ever subtracted again in the process?

(iii) Will this ever stop?

The key to this is to consider the number N and find the **largest F below N** and subtract that.

If N is an F already then just stop.

If N isn't an F how do I know I can find an F that is lower?

Well the first three counting numbers {1,2,3} are all Fs, so there is certainly an F that is lower. If there are several, then **pick the biggest!**

Let's call that F, 'b'. And while we are at it, call the one below that, 'a'.

Consider a + b.

It is the next F after b. Where does it come in relation to N?

Is it equal to N? If so, then N is an F and we can stop.

Is it below N? No, because we were told to pick the biggest F below N and call that one 'b'. So a + b cannot be under N.

So N < a + b => N - b < a

So the remainder from the subtraction is below 'a' and so there is no chance of ever repeating a subtraction of 'b'. The remainder is just too small. In fact, we can go further. Since N - b < a, that means that even 'a' cannot be subtracted from the remainder. So, having subtracted 'b', we can never subtract 'a' as well.

So that proves I can find an F and that I'll never repeat an F.

But will I ever stop?

Well the remainders get smaller and smaller whilst remaining positive. Eventually I'll have to reach an F because the last possible remainders, if no other F presents itself, will be 3 or 2 or 1 and they are all Fs.

So it is guaranteed that I will eventually end in an F.

Example:

N =150

What's the largest Fibonacci number under 150?

144 I think. That is 'b'. And 'a' = 89

Calculate N - b = 150 - 144 = 6. Notice this is under 89.

Replace 150 by 6.

N = 6. What is the largest F under this? Answer 5

Calculate N - b = 6 - 5 = 1

This is the new N. But** it is an F **so I can stop.

150 = 144 + 5 + 1

Note: The representation by a sum of Fs is not always unique (150 = 144 + 3 + 2 + 1) but choosing the largest each time is what guarantees that we don't repeat and we don't get two consecutive Fs.

Bob

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,525

Now that is simple. I understood it this time.

I have a little problem with your note in the end. Again, the Zeckendorf theorem states that every number has a **unique** representation as a sum of **non-adjacent** Fibonacci numbers. Your proof takes care of that, but your comment at the end ruins it. Just a little notice from me.

Here lies the reader who will never open this book. He is forever dead.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,393

Sure it does. But I wasn't trying to prove that theorem, just show the representation is possible. And that statement emphasises why the largest is needed at each stage. (challenge: Find a number with more than one representation that does satisfy the theorem.)

Thanks for reading it and allowing me the chance to show off some more.

Bob

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,525

Do you want me to find a number which can be represented as a sum of several non-adjacent F numbers in two different ways?

Here lies the reader who will never open this book. He is forever dead.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,393

Yes, if you can.

Bob

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,525

That is impossible. The impossibility is proven by the Zeckendorf theorem.

Here lies the reader who will never open this book. He is forever dead.

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Thank You,

Bob

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'Humanity is still kept intact. It remains within.' -Alokananda

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**Mark-nz****Member**- Registered: 2012-10-07
- Posts: 3

Thales, Archimedes, Eudoxus, Euclid, Diophantus, Newton are six pretty prominent guys.

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**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,532

I cannon believe no one suggested Gottfried Wilhelm Leibniz, possibly the greatest mathematician of all time and the original creator of Calculus.

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