# every sufficiently large even integer can be expressed as the sum of a pair of abundant numbers

###### Theorem 1.

If $n>1540539$, then $n=a+b$, where $a$ and $b$ are abundant numbers.

###### 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 $n=20m+k$ where $m$ and $k$ are positive and $k<20$. Note that, since $n>1540539$ and $k<20$, it follows that $m>77026>4054k$, hence we have

 $n=20(m-4054k)+81081k.$

Since positive multiples of abundant numbers are abundant, we may set $a=20(m-4054k)$ and $b=81081k$. ∎

Title every sufficiently large even integer can be expressed as the sum of a pair of abundant numbers EverySufficientlyLargeEvenIntegerCanBeExpressedAsTheSumOfAPairOfAbundantNumbers 2013-03-22 16:46:58 2013-03-22 16:46:58 rspuzio (6075) rspuzio (6075) 9 rspuzio (6075) Proof msc 11A05