In this puzzle, we asked: suppose and . What then is ?
First let us solve this by hand. Since , . So
Hence it follows that
But since , this means that , so if we must have or vice versa. In any case, it follows that .
The reason I wanted to present this problem is that it shows the power and beauty of Gröbner bases. Here is how to solve this problem in MAPLE. Define . Then the question is: what equation does satisfy? So we set up the three equations, two of which are the data of the problem and the last one is the definition of . Crank up MAPLE…
e1 := a+b-1:
e2 := a^3+b^3-1:
e3 := a^17+b^17-z:
[-b + b , a + b – 1, z – 1]
The Basis([e1,e2,e3],plex(z,a,b)) command computes the Gröbner basis of (the ideal defined by) this set of equations (with the right lexicographic ordering) and has to give a univariate polynomial in . The resulting three equations are equivalent (in terms of their zeroes) to the original ones. The only zero of is .
I leave many things unexplained on purpose: if you are interested, the help system of MAPLE and all the resources of the web are at your disposal.