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## #1 2012-09-10 01:08:33

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,826
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### Infinite Descent

The above equation has no integer solutions

Please teach me how to prove the above using the infinite descent

'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'
I'm not crazy, my mother had me tested.

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## #2 2012-09-10 01:32:09

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Infinite Descent

Hi Agnishom;

The proof is difficult.

http://fermatslasttheorem.blogspot.com/ … m-n-4.html

The proof requires the understanding of moduli. I can help with that if you want it.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #3 2012-09-11 16:26:06

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,826
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### Re: Infinite Descent

Yes Its quite complicated
See the last step .... it claims that they have reached infinite descent
How?

'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'
I'm not crazy, my mother had me tested.

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## #4 2012-09-11 16:44:51

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Infinite Descent

Hi Agnishom;

Step 8?

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #5 2012-09-12 00:18:02

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,826
Website

### Re: Infinite Descent

Yes.. that one

'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'
I'm not crazy, my mother had me tested.

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## #6 2012-09-12 00:59:44

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Infinite Descent

Hi;

To have an infinite descent in this case you have to prove that there is always integers like the ones you started with that have the same properties but are smaller. For any finite a,b you can not keep finding integers p,q that are smaller than a,b. You must run out.

Please read this which is explaining it much better than I can.

http://fermatslasttheorem.blogspot.com/ … scent.html

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #7 2012-09-13 01:35:51

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,826
Website

### Re: Infinite Descent

But now we have reached infinite descent since:
P^2 = a^2 + b^2 = p which is less than p^2 + q^2 = z which is less than z^2.

But how does this show that a and b will have similar properties as p & q??

'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'
I'm not crazy, my mother had me tested.

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## #8 2012-09-13 01:51:37

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Infinite Descent

Hi;

I think they are all squares. a^2 and b^2 and p^2 and q^2. You just have to be able to see that one group of squares is less than the other.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #9 2012-09-13 02:35:29

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,826
Website

### Re: Infinite Descent

So what?
They are squares... how does that show that there will be smaller numbers MATCHING THE ORIGINAL CONDITION?

'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'
I'm not crazy, my mother had me tested.

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## #10 2012-09-13 02:42:08

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Infinite Descent

Hi Agnishom;

There I am just as confused as you are. I do not have the whole proof understood either.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #11 2012-09-13 02:50:14

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,826
Website

### Re: Infinite Descent

Thats hard to believe
(Just joking)

Last edited by Agnishom (2012-09-13 02:51:23)

'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'
I'm not crazy, my mother had me tested.

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## #12 2012-09-13 02:58:46

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Infinite Descent

Hi;

Math is gigantic, bobbym is tiny. That explains it.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #13 2012-12-23 02:18:04

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,826
Website

### Re: Infinite Descent

bit.ly/infdescent
Isn't this one 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'
I'm not crazy, my mother had me tested.

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## #14 2012-12-23 02:20:50

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Infinite Descent

Hi;

Did you know that is the method to prove your latest triangle problem?

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #15 2012-12-23 03:13:06

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,826
Website

### Re: Infinite Descent

Pythagorean triangle?

'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'
I'm not crazy, my mother had me tested.

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## #16 2012-12-23 03:21:10

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Infinite Descent

Yes, that link solves the other problem.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #17 2012-12-23 03:22:04

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,826
Website

### Re: Infinite Descent

Okay, imagining the two legs are perfect squares, say a^2 and b^2
We can't have a hypotenuse c such that a^4 + b^4 = c^2

But what if we are talking about one leg and one hypotenuse?

'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'
I'm not crazy, my mother had me tested.

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## #18 2012-12-23 03:27:13

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Infinite Descent

Hi Agnishom;

I do not know about that one yet.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #19 2012-12-24 14:40:33

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,826
Website

### Re: Infinite Descent

Pleae explain me this: For Euclidean Integers, relatively prime divisors of n-powers are themselves n-powers.

'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'
I'm not crazy, my mother had me tested.

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## #20 2012-12-24 20:57:21

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,037

### Re: Infinite Descent

I do not understand the question... To have relatively prime numbers, you need two numbers.

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
The knowledge of some things as a function of age is a delta function.

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## #21 2012-12-25 02:10:14

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,826
Website

### Re: Infinite Descent

Suppose, w and v are relatively prime to each other
and

The theorem claims that there exists integers x and y such that
and

Example, 10^2 = 25*4 where (25, 4) are relatively prime to each other now see that 25 = 5^2 and 4 = 2^2

'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'
I'm not crazy, my mother had me tested.

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## #22 2012-12-25 06:36:33

scientia
Member
Registered: 2009-11-13
Posts: 224

### Re: Infinite Descent

Agnishom wrote:

Suppose, w and v are relatively prime to each other
and

The theorem claims that there exists integers x and y such that
and

Example, 10^2 = 25*4 where (25, 4) are relatively prime to each other now see that 25 = 5^2 and 4 = 2^2

If
, take
and
. Otherwise, since
and
are coprime, each prime divisor of
must divide
exactly as many times as it divides
, namely a multiple of
times. Thus
, being a product of primes of power a multiple of
, is a power of
. Same for
.

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## #23 2012-12-25 20:08:15

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,826
Website

### Re: Infinite Descent

Are you sure the proof is as simple as that?

'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'
I'm not crazy, my mother had me tested.

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## #24 2012-12-25 21:00:58

scientia
Member
Registered: 2009-11-13
Posts: 224

### Re: Infinite Descent

Do you understand it?

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## #25 2012-12-26 15:13:14

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,826
Website

### Re: Infinite Descent

I still have confusions, though it looks intutively obvious

'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'
I'm not crazy, my mother had me tested.

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