### What a difference a minus one makes!

The FLT - Fermat's Last Theorem - states that for n ≥ 3, the Diophantine equation

^{n}+ y

^{n}= z

^{n}

has no solutions. We now know this is true due to the 1995 result of Andrew Wiles. The theorem was conjectured by Pierre de Fermat in 1637 and thus withstood the many attempts during the 350 years' history.

There exist several slight modifictions that make the equation amenable to the efforts of high school students.

I have recently learned of one related modification that actually reverses the result. To boot, the equation

x^{ n} + y^{ n} = z^{ n-1}

has has infinitely many solutions for all n ≥ 3. Just a measly minus 1. But what a difference it makes!

There is more to that theme. In an answer to a question posted by Morrey Klamkin Leo Moser gave a solution to a more general equation

x^{ a} + y^{ b} = z^{ c} ,

where (a, b, c) = 1, i.e., where a, b, c are mutually prime.