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## #1 2014-05-09 06:38:48

engrymbiff
Member
Registered: 2010-06-14
Posts: 30

### A number theory problem

Assume that x and y are whole numbers and prove that

is satisfied by w and z (where w and z also is whole numbers) if and only if the product of x and y is even.

Last edited by engrymbiff (2014-05-11 04:40:51)

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## #2 2014-05-09 06:55:44

bob bundy
Moderator
Registered: 2010-06-20
Posts: 7,979

### Re: A number theory problem

hi engrymbiff

has roots w and z

What variable(s) are we talking about here?

eg.  in

the roots for x are 2 and 3

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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## #3 2014-05-11 04:42:12

engrymbiff
Member
Registered: 2010-06-14
Posts: 30

### Re: A number theory problem

bob bundy wrote:

hi engrymbiff

has roots w and z

What variable(s) are we talking about here?

eg.  in

the roots for x are 2 and 3

Bob

Sorry for my sloppy formulation, I've reformulated the problem (see above).

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## #4 2014-05-11 05:12:06

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

### Re: A number theory problem

Hi;

This might help:

The question can rephrased as

so when is the sum of 3 squares a square?

Using the identity

We can say that x = (n+1), y = (n(n+1)) and w = n.

When n is odd then it obvious that both x and y are even so xy is even. When n is even y is even and x is odd but xy is still even. The only thing I can not prove that all solutions are of the form given in the identity.

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 2014-05-11 05:20:04

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,775
Website

### Re: A number theory problem

Much like the other days problem..

'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 2014-05-11 05:25:29

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

### Re: A number theory problem

In what way?

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 2014-05-11 05:40:03

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,775
Website

### Re: A number theory problem

In the way that you've proved p => q when you were asked to prove q => p

'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 2014-05-11 05:42:33

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

### Re: A number theory problem

I did not say it was a proof but it does work for the one identity posted. I can not prove the z^2 is always of the form of the RHS of the identity.

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 2014-05-11 05:43:49

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,775
Website

### Re: A number theory problem

Sorry, I hope I did not offend you...

'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 2014-05-11 05:50:57

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

### Re: A number theory problem

Kaboobly doo! Who said anything about offending me? Who cares about that anyhow. People call me stupid all the time, it does not bother me. Now if they forget to call me for ice cream that does bother me.

Is the idea okay for the example provided?

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 2014-05-11 05:54:24

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,775
Website

### Re: A number theory problem

I don't think so.

You're assuming this out of the blue:

bobbym wrote:

We can say that x = (n+1), y = (n(n+1)) and w = n.

What if there is another kind of x,y,w that satisfies the 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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## #12 2014-05-11 05:55:45

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,775
Website

### Re: A number theory problem

I do.

MIF wrote:

Be respectful of Administrators and Moderators as they have to make difficult decisions on behalf of all members.

'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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## #13 2014-05-11 05:59:35

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

### Re: A number theory problem

I wrote that.

Call me a dummy and I will argue with you, eat my ice cream and you are dead.

What I have tried to prove for that identity is that xy is always even but just for that identity.

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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## #14 2014-05-11 06:05:39

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,775
Website

### Re: A number theory problem

You wrote that? LOL!!

Do not get me wrong, I do not think you are a dummy. However, I shall argue with you even if I like your ideas because I like argument more.

On which forum is your ice cream available?

'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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## #15 2014-05-11 06:07:16

ShivamS
Member
Registered: 2011-02-07
Posts: 3,648

### Re: A number theory problem

I did not know 93 year old people can even eat ice cream.

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## #16 2014-05-11 06:08:35

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

### Re: A number theory problem

When you are 93 you will eat ice cream.

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 2014-05-11 12:47:19

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

### Re: A number theory problem

Hi engrymbiff;

This is the best I can find for you. We can improve on post #4.

The question can rephrased as

so when is the sum of 3 squares a square?

Just as there are Pythagorean triples these are called Pythagorean Quadruples. it has been proven by Oliverio 1996 that only when x and y are positive and even, then w and z as integers will exist. When x and y are positive and odd ( a necessary condition so that xy is not even ) then w and z as integers do not exist.

http://archive.lib.msu.edu/crcmath/math/math/p/p745.htm

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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## #18 2014-05-11 13:18:50

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,775
Website

### Re: A number theory problem

That page has got no proofs in it

'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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## #19 2014-05-11 13:20:59

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

### Re: A number theory problem

Yes, I know But the same result has been posted elsewhere. I am looking for the Oliverio paper online.

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.

Offline