# 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 |