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 March 8th, 2012, 08:47 PM #1 Senior Member   Joined: Jan 2009 Posts: 344 Thanks: 3 Find the largest value for which the system has a solution For positive integer values of m and n find the largest value for which the system; $a+b=m$ $a^2+b^2=n$ $a^3+b^3=m+n$ has a solution.
March 9th, 2012, 04:27 AM   #2
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Re: Find the largest value for which the system has a soluti

Hello, sivela!

I have a start on this problem . . . Can anyone finish it?

Quote:
 For positive integers $m$ and $n$, find the largest values for which this system has a solution; [color=beige]. . [/color]$\begin{Bmatrix}a\,+\,b=&m=&[1] \\ \\ \\ a^2\,+\,b^2=&n=&[2] \\ \\ \\ a^3\,+\,b^3=&m+n=&[3]\end{Matrix}=$

$\text{Square [1]: }\:(a\,+\,b)^2 \:=\:m^2 \;\;\;\Rightarrow\;\;\;\underbrace{a^2\,+\,b^2}_{\ text{This is }n}\,+\,2ab \:=\:m^2$

[color=beige]. . . . . . . . . [/color]$n\,+\,2ab \:=\:m^2 \;\;\;\Rightarrow\;\;\;ab \:=\:\frac{m^2\,-\,n}{ab}$

$\text{Cube [1]: }\:(a\,+\,b)^3 \:=\:m^3 \;\;\;\Rightarrow\;\;\;a^3\,+\,3a^2b\,+\,3ab^2\,+\ ,b^3 \:=\:m^3$

[color=beige]. . . . . . [/color]$\underbrace{a^3\,+\,b^3}_{m+n} \,+\,3\underbrace{ab}_{\frac{m^2-n}{2}}\underbrace{(a\,+\,b)}_{m} \:=\:m^3$

[color=beige]. . . . . . [/color]$m\,+\,n\,+\,3\left(\frac{m^2\,-\,n}{2}\right)m \;=\;m^3$

[color=beige]. . . . . . [/color]$m^3\,-\,3mn\,+\,2m\,+\,2n \:=\:0$

Now what?

 March 9th, 2012, 09:27 AM #3 Senior Member   Joined: Feb 2012 Posts: 628 Thanks: 1 Re: Find the largest value for which the system has a soluti We have that $a^3 - a^2 - a + b^3 - b^2 - b= 0$; $a(a^2 - a - 1) + b(b^2 - b - 1)= 0$, which we observe is true if both $a^2 - a - 1$ and $b^2 - b - 1$ are 0. Using this observation, we get the solution $a= \frac{1 + \sqrt{5}}{2}, b = \frac{1 - \sqrt{5}}{2}$, from which $a + b= 1$ $a^2 + b^2= 3$ $a^3 + b^3= 4$. I wonder if there are other solutions...

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