11A05-EverySufficientlyLargeEvenIntegerCanBeExpressedAsTheSumOfAPairOfAbundantNumbers.tex

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\pmtitle{every sufficiently large even integer can be expressed as the sum of a pair of abundant numbers}
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\pmtype{Proof}
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\begin{document}
\begin{thm}
If $n > 1540539$, then $n = a + b$, where $a$ and
$b$ are abundant numbers.
\end{thm}

\begin{proof}
Note that both $20$ and $81081$ are abundant numbers.
Furthermore, we have $81081 = 4054 \cdot 20 + 1$.
If $n$ is a multiple of $20$, then $n-20$ is also 
a multiple of $20$ hence, as a multiple of an abundant
number, is also abundant, so we may choose $a = 20$ 
and $b = n-20$.  Otherwise, write $n = 20 m + k$ where
$m$ and $k$ are positive and $k < 20$.  Note that,
since $n > 1540539$ and $k < 20$, it follows that
$m > 77026 > 4054 k$, hence we have
\[
n = 20 (m - 4054 k) + 81081 k.
\]  
Since positive multiples of abundant numbers are 
abundant, we may set $a = 20 (m - 4054 k)$ and
$b = 81081 k$.
\end{proof}

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