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**Mario23****Member**- Registered: 2017-01-03
- Posts: 20

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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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 108,518

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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**Mario23****Member**- Registered: 2017-01-03
- Posts: 20

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

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 108,518

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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**Mario23****Member**- Registered: 2017-01-03
- Posts: 20

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

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**Mathegocart****Member**- Registered: 2012-04-29
- Posts: 1,858

Here's a crucial hint to prove there is only one solution: note the essential fact that if

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

The integral of hope is reality.

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**Mario23****Member**- Registered: 2017-01-03
- Posts: 20

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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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 108,518

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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**Mario23****Member**- Registered: 2017-01-03
- Posts: 20

Okay,I am looking forward to seeing your ideas

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 108,518

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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**Mario23****Member**- Registered: 2017-01-03
- Posts: 20

Thank you for your help !

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 108,518

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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**Mario23****Member**- Registered: 2017-01-03
- Posts: 20

They are right,I checked them

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

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 108,518

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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