Quandaries
and Queries 

Hello I am a lecturer and am having a problem with the following Proof by If (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 Peter 

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 

