# theorem on multiples of abundant numbers

Theorem. The product $nm$ of an abundant number $n$ and any integer $m>0$ is also an abundant number, regardless of the abundance or deficiency^{} of $m$.

Proof. Choose an abundant number $n$ with $k$ divisors^{} ${d}_{1},\mathrm{\dots},{d}_{k}$ (where the divisors are sorted in ascending order and ${d}_{1}=1$, ${d}_{k}=n$) that add up to $2n+a$, where $a>0$ is the abundance of $n$. For maximum flair, set $a=1$, the bare minimum for abundance (that is, a quasiperfect number). Next, for $m$ choose a spectacularly deficient number such that $\mathrm{gcd}(m,n)=1$, preferably some large prime number. If we choose a prime number^{}, its divisors will only add up to $m+1$. However, the divisors of $nm$ will include each ${d}_{i}m$, where ${d}_{i}$ is a divisor of $n$ and $$. Therefore, the divisors of $nm$ will add up to

$$\sum _{i=1}^{k}{d}_{i}+\sum _{i=1}^{k}{d}_{i}m=2nm+a(m+1)+2n.$$ |

It now becomes obvious that by insisting that $m$ and $n$ be coprime^{} we are guaranteeing that if $m$ is itself prime, it will bring at least $k$ new divisors to the table. But what if $\mathrm{gcd}(m,n)>1$, or in the most extreme case, $m=n$? In such a case, we just can’t use the same formula for the sum of divisors of $nm$ that we used when $m$ and $n$ were coprime, as that would count some divisors twice. However, $m=n$ still brings new divisors to the table, and those new divisors add up to

$$\sum _{i=2}^{k}{d}_{i}{d}_{k}=2{n}^{2}+2{a}^{2}+a.$$ |

Having proven these extreme cases, it is obvious that $nm$ will be abundant in other cases, such as $m$ being a composite deficient number, a perfect number, an abundant number sharing some but not all prime factors^{} with $n$, etc.

Title | theorem on multiples of abundant numbers |
---|---|

Canonical name | TheoremOnMultiplesOfAbundantNumbers |

Date of creation | 2013-03-22 16:05:46 |

Last modified on | 2013-03-22 16:05:46 |

Owner | CompositeFan (12809) |

Last modified by | CompositeFan (12809) |

Numerical id | 15 |

Author | CompositeFan (12809) |

Entry type | Theorem |

Classification | msc 11A05 |

Related topic | APositiveMultipleOfAnAbundantNumberIsAbundant |