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 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 . 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 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
Canonical name	EverySufficientlyLargeEvenIntegerCanBeExpressedAsTheSumOfAPairOfAbundantNumbers
Date of creation	2013-03-22 16:46:58
Last modified on	2013-03-22 16:46:58
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	9
Author	rspuzio (6075)
Entry type	Proof
Classification	msc 11A05