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© 1999-2010 by Francis Edward Su

From the Fun Fact files, here is a Fun Fact at the Easy level:

# tamreF's Last Theorem

Here's an amusing theorem that is very easy to prove, which we'll call tamreF's Last Theorem. In fact it is tamreF's only known theorem. It says:

The following equation

Nx + Ny = Nz
has no solutions in positive integers for N greater than 2.

This theorem has not been unsolved for centuries; in fact, it was solved as soon as it was posed. It is much easier to prove than its counterpart: Fermat's Last Theorem.

The Math Behind the Fact:
The proof is ridiculously easy; consider the equation in base N. Perfect powers of N consist of a 1 followed by many zeroes. Adding two such numbers cannot be a number of that form unless N=2 and the 1's are in the same position.

Su, Francis E., et al. "tamreF's Last Theorem." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

References:
I have been told that the reference is in a Martin Gardner book, but I don't have the reference!

Keywords:    Fermat's last theorem
Subjects:    number theory
Level:    Easy
Fun Fact suggested by:   Arthur Benjamin
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