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Solve for m, n, s and t

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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,678
Hi MHB,

Problem:

Solve for reals of

$m-n-s+t=1$

$m^2+n^2-s^2-t^2=3$

$m^3-n^3-s^3+t^3=-5$

$m^4+n^4-s^4-t^4=15$

I've encountered this problem a while back and I've tried to use many methods (which include by manipulating some inequality theorems or solving them by the elimination of variables method or trying to relate the second equation and the third by multiplying the second and the third (after changing the minus sign) and let it equal to the 4th equation but all these methods have fallen apart. I am getting very tired of it and hence I hope someone could help me by giving me some hints so that I can finish the unfinished problem.

Thanks.
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,774
Hi MHB,

Problem:

Solve for reals of

$m-n-s+t=1$

$m^2+n^2-s^2-t^2=3$

$m^3-n^3-s^3+t^3=-5$

$m^4+n^4-s^4-t^4=15$

I've encountered this problem a while back and I've tried to use many methods (which include by manipulating some inequality theorems or solving them by the elimination of variables method or trying to relate the second equation and the third by multiplying the second and the third (after changing the minus sign) and let it equal to the 4th equation but all these methods have fallen apart. I am getting very tired of it and hence I hope someone could help me by giving me some hints so that I can finish the unfinished problem.

Thanks.
I was looking at $m^4+n^4-s^4-t^4=15$ and wondered...
Could I find a solution for just this equation without going to great lengths?

Well... it might be something like $(\pm 2)^4$ combined with a couple of $(\pm 1)^4$.

And what do you know... it fits! ;)
 
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  • #3

anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,678
Hi I like Serena, thank you very much for your reply.

You're absolutely right because based on your observation, we can tell that $(-2, -1, -1, 1)$ and $(1, 2, -1, 1)$ are two credible answers to the problem and now, I think the remaining effort is to show that these are the only two possible answers. (Smile)