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Title: integer Post by tony123 on Sep 1st, 2007, 1:31am if x is integer find all integer x such that 9x^2+160x+800 is a perfect squer |
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Title: Re: integer Post by towr on Sep 2nd, 2007, 7:06am There are infinitely many, I reckon. Do you want a recurrence or a closed formula to represent them all? |
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Title: Re: integer Post by srn347 on Sep 3rd, 2007, 11:54am Is i an integer? |
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Title: Re: integer Post by towr on Sep 3rd, 2007, 12:16pm on 09/03/07 at 11:54:03, srn347 wrote:
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Title: Re: integer Post by Sameer on Sep 3rd, 2007, 12:53pm Here's a start [hide] 9x2 + 160x + 800 = 9x2 + 6ax + a2 + bx + bc Where 6ax + bx = 160x or 6a + b = 0 (since x cannot be 0) a2 + bc = 800 Giving (3x + a)2 + b(x + c) This is a perfect square when x = -c (b cannot be 0) I am stuck here!! [/hide] |
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Title: Re: integer Post by towr on Sep 3rd, 2007, 1:19pm A few that are easy to find: [hide]-31, -14, -7, -5, 2[/hide] The next one is surprisingly far away (up to the point I really need a large integer library to be sure it even is one). It's rather contrary to my experience with these kinds of problems. |
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Title: Re: integer Post by Eigenray on Sep 3rd, 2007, 1:49pm on 09/03/07 at 13:19:51, towr wrote:
Without using a large integer library, I can be sure that [hide]it isn't[/hide]. |
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Title: Re: integer Post by Sameer on Sep 3rd, 2007, 3:04pm I got the [hide] 2 and -5 [/hide] without resorting to a program. I was wondering if we can get it analytically!! |
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Title: Re: integer Post by Eigenray on Sep 4th, 2007, 8:44pm on 09/03/07 at 15:04:45, Sameer wrote:
Did you try completing the square? |
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Title: Re: integer Post by Sameer on Sep 5th, 2007, 8:33am on 09/04/07 at 20:44:39, Eigenray wrote:
I didn't follow that!! |
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Title: Re: integer Post by Eigenray on Sep 5th, 2007, 10:15am [link=http://en.wikipedia.org/wiki/Completing_the_square]Completing the square[/link]: 9x^2+160x+800 = (3x + 80/3)2 + 800/9. It is the fundamental trick of quadratic expressions. |
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