Fermat's Last Theorem - The Search for a Generalization
One elegant approach is to write the theorem in the language of metric spaces..
Let (x1,x2,..,xn) be a point in a n-dimensional vector space.
Then r^q=|x1|^q+|x2|^q+...+|xn|^q defines a series of metrics for q=1,2,3,.. where r is the distance of the point from the origin and |xi| is the absolute value of xi.
This allows a generalized version of Fermat's Last Theorem to be written as follows..
"An integer point (x1,x2,..,xn) is never an integer distance (r) from the origin when q>n"
It's interesting to note that the generalization holds when the metric parameter (q) exceeds the dimensionality of the space (n).
The special case of n=2 was proved in 1994 by Andrew Wiles. It was an amazing achievement because mathematicians had been trying to prove it since Fermat first suggested it in 1637.
But is this generalized version true?
At first I though it was, but one of my readers pointed out a counter example. And one counter example is all you need to disprove a conjecture! However, I am now investigating ways the conjecture may be modified (or even generalized further) so it still holds.
Content written and posted by Ken Abbott firstname.lastname@example.org