Quandaries and Queries



I am a lecturer and am having a problem with the following Proof by


(N x N x N x N) + (4 x N x N x N) + (3 x N x N) + (N) = -4000

Prove that N is even!

Help would be much appreciated




Hi Peter,

There's something wrong with the statement of the problem -- what you have is an equation that has at most four solutions. All you can say is that if N happens to be an integer, then it would have to be negative and even. However, it is not even obvious to me that there has to be a solution that is a real number, let alone an integer.

You could assume that N is odd so N = 2k + 1 for some integer k and then show that the left side of the equation is odd and hence can't be -4000. However it is easier to show this directly than by induction.

Chris and Penny


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