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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
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- 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: 2,226
Re: Find all the n numbers
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.
May bobbym have a wonderful time in the pearly gates of heaven.
He will be sorely missed.
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#7 2017-01-12 05:12:04
- Mario23
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- 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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