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## #1 2017-01-11 07:17:28

Mario23
Member
Registered: 2017-01-03
Posts: 27

### Find all the n numbers

Hey! I've come across an exercise I can 't seem to solve.
Find all the n integers that satisfy :

is a rational number.
I can't see how to find them because 4n-2 can't be a perfect square and that's why I need some help on it.

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## #2 2017-01-11 07:46:51

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

### Re: Find all the n numbers

Hi;

n = 13 can be found quickly.

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 2017-01-11 08:18:42

Mario23
Member
Registered: 2017-01-03
Posts: 27

### Re: Find all the n numbers

Yes,I managed to guess it too but I don't know how to find all of them.

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## #4 2017-01-11 10:42:09

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

### Re: Find all the n numbers

What makes you think there are more?

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 2017-01-11 17:36:53

Mario23
Member
Registered: 2017-01-03
Posts: 27

### Re: Find all the n numbers

Well I don't know.There are usually more .And I don't know how to prove there aren't.

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## #6 2017-01-11 22:44:54

Mathegocart
Member
Registered: 2012-04-29
Posts: 1,884

### Re: Find all the n numbers

Here's a crucial hint to prove there is only one solution: note the essential fact that if $\sqrt{p/q} \in Q, q\sqrt{p/q} \in Q$

Last edited by Mathegocart (2017-01-11 22:45:11)

The integral of hope is reality.

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## #7 2017-01-12 05:12:04

Mario23
Member
Registered: 2017-01-03
Posts: 27

### Re: Find all the n numbers

My best try at proving that using that relation was that

where m is a rational number.I used then that the other frac is mq and I tried to square them,divide the ecuations etc but it is definitely not working.

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## #8 2017-01-12 05:57:59

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

### Re: Find all the n numbers

Some ideas that I am working on seem to validate my view in post #4. 13 seems to be the only solution.

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 2017-01-12 06:42:35

Mario23
Member
Registered: 2017-01-03
Posts: 27

### Re: Find all the n numbers

Okay,I am looking forward to seeing your ideas

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## #10 2017-01-13 06:06:32

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

### Re: Find all the n numbers

We can say that

Solving simultaneously:

{{x = -5, y = -3}, {x = -5, y = 3}, {x = 5, y = -3}, {x = 5, y = 3}}

Are the only 4 solutions and they all give n = 13.

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 2017-01-14 06:00:57

Mario23
Member
Registered: 2017-01-03
Posts: 27

### Re: Find all the n numbers

Thank you for your help !

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## #12 2017-01-14 10:26:23

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

### Re: Find all the n numbers

Those were just thoughts, that might have a big hole in 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 2017-01-14 22:21:58

Mario23
Member
Registered: 2017-01-03
Posts: 27

### Re: Find all the n numbers

They are right,I checked them

Last edited by Mario23 (2017-01-14 22:22:22)

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## #14 2017-01-15 06:52:48

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

### Re: Find all the n numbers

Hi;

The solutions found in post #10 are correct but I am afraid their might be more that ht method I used there is missing. I hate using reasoning in math, that is why I dislike and mistrust the whole concept of proof. Trouble is, computation although more reliable can leave you hanging.

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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