sayhisam1
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| Joined: 25 Nov 2009 |
| Total Posts: 2092 |
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| 17 Aug 2016 01:36 AM |
| Prove that if (a^2 + b^2) / (ab + 1) is integer, then it is a perfect square. |
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Laedere
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| Joined: 17 Jun 2013 |
| Total Posts: 23601 |
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| 17 Aug 2016 01:42 AM |
how does one prove a formula
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pidgey
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| Joined: 16 Jan 2008 |
| Total Posts: 2739 |
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| 17 Aug 2016 01:43 AM |
Rearrange to get a2−akb+b2−k=0a2−akb+b2−k=0, as a quadratic in aa this has two values: aa and kb−a=(b2−k)/akb−a=(b2−k)/a. (The second root is determined in two different ways from the expansion (x−r1)(x−r2)=x2−(r1+r2)x+r1r2(x−r1)(x−r2)=x2−(r1+r2)x+r1r2.)
Now suppose we have a,ba,b such that kk is an integer but not a square, by the investigation about roots we have that the second root is a nonzero integer since k,b,ak,b,a are integers and k≠b2k≠b2, futhermore it is positive which is easily seen from its defining equation.
WLOG assume a≥ba≥b so that the second root is strictly smaller than aa. This leads to a decent, replacing aa with the second root.
https://www.roblox.com/My/Groups.aspx?gid=2893266 | pidgey is my name sh##posting is my game https://www.roblox.com/money-item?id=13127780 |
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sayhisam1
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| Joined: 25 Nov 2009 |
| Total Posts: 2092 |
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| 17 Aug 2016 01:45 AM |
| ^Congrats, you know how to copy and paste. |
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pidgey
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| Joined: 16 Jan 2008 |
| Total Posts: 2739 |
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| 17 Aug 2016 01:55 AM |
you too?
https://www.roblox.com/My/Groups.aspx?gid=2893266 | pidgey is my name sh##posting is my game https://www.roblox.com/money-item?id=13127780 |
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Eddimor
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| Joined: 23 Aug 2011 |
| Total Posts: 294 |
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pidgey
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| Joined: 16 Jan 2008 |
| Total Posts: 2739 |
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| 17 Aug 2016 02:03 AM |
actually the answer is 42
https://www.roblox.com/My/Groups.aspx?gid=2893266 | pidgey is my name sh##posting is my game https://www.roblox.com/money-item?id=13127780 |
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Eddimor
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| Joined: 23 Aug 2011 |
| Total Posts: 294 |
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| 17 Aug 2016 02:05 AM |
Pies in France cost 49 Euros, so I'm basically right.
-Edd |
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MiniNob
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| Joined: 14 May 2013 |
| Total Posts: 822 |
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| 17 Aug 2016 02:44 AM |
| now let me ask you is you know what a descent is |
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