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Hard math problem for you all

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sayhisam1 is not online. sayhisam1
Joined: 25 Nov 2009
Total Posts: 2092
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 is not online. Laedere
Joined: 17 Jun 2013
Total Posts: 23601
17 Aug 2016 01:42 AM
how does one prove a formula


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pidgey is not online. pidgey
Joined: 16 Jan 2008
Total Posts: 2739
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.


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sayhisam1 is not online. sayhisam1
Joined: 25 Nov 2009
Total Posts: 2092
17 Aug 2016 01:45 AM
^Congrats, you know how to copy and paste.
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pidgey is not online. pidgey
Joined: 16 Jan 2008
Total Posts: 2739
17 Aug 2016 01:55 AM
you too?


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Eddimor is not online. Eddimor
Joined: 23 Aug 2011
Total Posts: 294
17 Aug 2016 02:02 AM
The answer is Pie.


-Edd
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pidgey is not online. pidgey
Joined: 16 Jan 2008
Total Posts: 2739
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 is not online. Eddimor
Joined: 23 Aug 2011
Total Posts: 294
17 Aug 2016 02:05 AM
Pies in France cost 49 Euros, so I'm basically right.


-Edd
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MiniNob is not online. MiniNob
Joined: 14 May 2013
Total Posts: 822
17 Aug 2016 02:44 AM
now let me ask you is you know what a descent is
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