determination of abundant numbers with specified prime factors
The formula for sums of factors may be used to find all abundant numbers with a specified set of prime factors or that no such numbers exist. To accomplish this, we first do a little algebraic manipulation to our formula.
A number whose factorization into prime numbers is is abundant if and only if
Note that each of the terms in the product is bigger than 1. Furthemore, the -th term is bounded by
This means that it is only possible to have an abundant number whose prime factors are if
As it turns out, the convers also holds, so we have a nice criterion for determining when a set of prime numbers happens to be the set of prime divisors of an abundant number.
A finite set of prime numbers is the set of prime divisors of an abundant number if and only if
As described above, if is a set of prime factors of an abundant number, then we may bound each term in the inequality of the previous theorem to obtain the inequality in the current theorem. Assume, then that is a finite set of prime numbers which satisfies said inequality. Then, by continuity, there must exist a real number such that
whenver . Since when , we can, for every , find an such that
so, by the previous theorem, is an abundant number. ∎
|Title||determination of abundant numbers with specified prime factors|
|Date of creation||2013-03-22 16:47:41|
|Last modified on||2013-03-22 16:47:41|
|Last modified by||rspuzio (6075)|