Ich Spreche Deutsch!

The other day was successful for two reasons.

(1) I went to the bank to make a transfer.  AND I spoke entirely in German!  Usually I am too afraid to do things all in German that involve my bank, but I did it!

(2) I proved a claim that Herbert and I had been trying to prove.  We had a function, say f(x, k, l), and wanted to find the zeros for fixed k,l.  Originally, we thought that there were three, but then we realized there was only one zero after plotting the function for certain values of k, l.  For k+l odd, we found a nice closed form for the zeros (about sqrt(k+l)).  And, the numbers we found suggested that this is the solution for all k,l.  So, I tried to put it into Mathematica to verify that this is the solution.  For odd k+l, it outputted 0.  Perfect.  But, for even k+l it outputted a complicated expression.  After looking at the expression for a while, I realized that this too was equal to zero.  Now, I needed to generalize this for all k,l, not just for the few examples that I tried out.  So, I tried varying the values of k,l to find a pattern in the complicated expression.  I first found a pattern for k=0.  Then, I was able to prove that f(~sqrt(k+l,k,l)=0 for k=0!  It turned out that I could easily add k back into the equation to prove the result for all k,l (even non-integers).  🙂  So, I proved something that we had been looking at due to Mathematica’s inability to recognize that a complicated expression is equal to zero.  Perhaps a more elegant way of explaining it is that I proved the claim by first proving it for a few examples and then generalizing the proof.

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