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\noindent {\large \bf \textsf{A Consequence of the Pythagorean Theorem}}\\


\noindent The well known Pythagorean theorem $a^2 + b^2 = c^2$ relates the lengths of the three sides of a right triangle. We present an exciting consequence of this result, namely, that if two of the three sides of a right triangle have lengths that are even integers, then the third side must be of even length as well. 

To see this, suppose that $a$ and $b$ are even; then $a = 2a'$, $b = 2b'$, where $a'$ and $b'$ are positive integers.  Hence
\[   c^2 = (2a')^2 + (2b')^2  = 4(a'^2 + b'^2).     \]
The result when the hypotenuse and one of the legs are of even length is proved similarly.

\begin{thebibliography}{1}
\bibitem{Ing} Ingham, A. E. (1932). \textit{The Distribution of Prime Numbers}. Cambridge: Cambridge University Press.
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\rightline{---Submitted by Carl F.\ Gauss}


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\footnotesize{doi.org/10.XXXX/amer.math.monthly.122.XX.XXX}

\footnotesize{MSC: Primary 00X00, Secondary 11Y11; 22Z22}

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