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Topic: integer (Read 398 times) |
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tony123
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if x is integer
find all integer x
such that
9x^2+160x+800
is a perfect squer
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towr
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Re: integer
« Reply #1 on: Sep 2nd, 2007, 7:06am » |
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There are infinitely many, I reckon.
Do you want a recurrence or a closed formula to represent them all?
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srn437
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Re: integer
« Reply #2 on: Sep 3rd, 2007, 11:54am » |
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Is i an integer?
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towr
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Re: integer
« Reply #3 on: Sep 3rd, 2007, 12:16pm » |
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on Sep 3rd, 2007, 11:54am, srn347 wrote:It's a gaussian integer, not a regular one though.
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Sameer
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Re: integer
« Reply #4 on: Sep 3rd, 2007, 12:53pm » |
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Here's a start
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!!
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"Obvious" is the most dangerous word in mathematics.
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towr
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Re: integer
« Reply #5 on: Sep 3rd, 2007, 1:19pm » |
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A few that are easy to find: -31, -14, -7, -5, 2
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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Eigenray
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Re: integer
« Reply #6 on: Sep 3rd, 2007, 1:49pm » |
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on Sep 3rd, 2007, 1:19pm, towr wrote:| 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). |
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Without using a large integer library, I can be sure that it isn't.
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Sameer
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Re: integer
« Reply #7 on: Sep 3rd, 2007, 3:04pm » |
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I got the 2 and -5 without resorting to a program. I was wondering if we can get it analytically!!
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"Obvious" is the most dangerous word in mathematics.
--Bell, Eric Temple
Proof is an idol before which the mathematician tortures himself.
Sir Arthur Eddington, quoted in Bridges to Infinity
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Eigenray
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Re: integer
« Reply #8 on: Sep 4th, 2007, 8:44pm » |
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on Sep 3rd, 2007, 3:04pm, Sameer wrote:| I got the 2 and -5 without resorting to a program. I was wondering if we can get it analytically!! |
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Did you try completing the square?
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Sameer
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Re: integer
« Reply #9 on: Sep 5th, 2007, 8:33am » |
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on Sep 4th, 2007, 8:44pm, Eigenray wrote:
Did you try completing the square? |
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I didn't follow that!!
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"Obvious" is the most dangerous word in mathematics.
--Bell, Eric Temple
Proof is an idol before which the mathematician tortures himself.
Sir Arthur Eddington, quoted in Bridges to Infinity
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Eigenray
wu::riddles Moderator
Uberpuzzler
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Posts: 1948
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Re: integer
« Reply #10 on: Sep 5th, 2007, 10:15am » |
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Completing the square:
9x^2+160x+800 = (3x + 80/3)2 + 800/9.
It is the fundamental trick of quadratic expressions.
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