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\noindent {\bf Problem statement} About a decade ago three
centuries of effort by mathematicians culminated in a proof that there
were no solutions to the Fermat equation
$$a^n+b^n = c^n$$ \noindent if $a$, $b$, $c$, and $n$ are positive
integers, with $n >2$.  There are, of course, solutions when $n=2$: for
example, $3^2+4^2=5^2$.

\medskip

\noindent a) Does the equation 
$$4^x + 5^x = 6^x$$ \noindent have any solution? (The word
``integer'' does not appear in the preceding sentence!) If there is a
solution, find an approximate value of this solution with accuracy
$\pm .05$. If there is no solution, explain why.

\medskip

\noindent b) Suppose $a$, $b$, and $c$ are positive real
numbers. Explore whether the equation $$a^x+b^x=c^x$$ \noindent must
have a solution.  This is a ``free form'' question: try to answer it
as well as you can. You are not asked to provide a ``formula'' for
$x$. You are asked to find conditions which will guarantee that such
an $x$ either does or does not exist.

\vfil\eject\end

