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riddles >> medium >> integer
(Message started by: tony123 on Sep 1st, 2007, 1:31am)

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

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?

Title: Re:  integer
Post by srn347 on Sep 3rd, 2007, 11:54am
Is i an integer?

Title: Re:  integer
Post by towr on Sep 3rd, 2007, 12:16pm

on 09/03/07 at 11:54:03, srn347 wrote:
Is i an integer?
It's a gaussian integer (http://en.wikipedia.org/wiki/Gaussian_integer), not a regular one though.

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]

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.

Title: Re:  integer
Post by Eigenray on Sep 3rd, 2007, 1:49pm

on 09/03/07 at 13:19:51, 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).

Without using a large integer library, I can be sure that [hide]it isn't[/hide].

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!!

Title: Re:  integer
Post by Eigenray on Sep 4th, 2007, 8:44pm

on 09/03/07 at 15:04:45, Sameer wrote:
I got the [hide] 2 and -5 [/hide] without resorting to a program. I was wondering if we can get it analytically!!

Did you try completing the square?

Title: Re:  integer
Post by Sameer on Sep 5th, 2007, 8:33am

on 09/04/07 at 20:44:39, Eigenray wrote:
Did you try completing the square?

I didn't follow that!!

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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