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

Theorem 1.

If $n\mathrm{>}\mathrm{1540539}$, then $n\mathrm{=}a\mathrm{+}b$, 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 alsoa multiple of $20$ hence, as a multiple of an abundantnumber, 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 . Note that,since $n>1540539$ and , it follows that$m>77026>4054k$, hence we have

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

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