every sufficiently large even integer can be expressed as the sum of a pair of abundant numbers
If , then , where and are abundant numbers.
Note that both and are abundant numbers. Furthermore, we have . If is a multiple of , then is also a multiple of hence, as a multiple of an abundant number, is also abundant, so we may choose and . Otherwise, write where and are positive and . Note that, since and , it follows that , hence we have
Since positive multiples of abundant numbers are abundant, we may set and . ∎
|Title||every sufficiently large even integer can be expressed as the sum of a pair of abundant numbers|
|Date of creation||2013-03-22 16:46:58|
|Last modified on||2013-03-22 16:46:58|
|Last modified by||rspuzio (6075)|