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.
Theorem 1.
A number whose factorization
into prime numbers is
is abundant if and only if
Proof.
By definition is abundant, if the sum of
the proper divisors of is greater than .
Using our formula, this is equivalent to the condition
Dividing the -th term in the product on
the left-hand side by the -th term on the
right-hand side,
so the condition becomes
∎
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.
Theorem 2.
A finite set of prime numbers is the set of
prime divisors of an abundant number if and only if
Proof.
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
Hence,
so, by the previous theorem, is an abundant number. ∎
Title | determination of abundant numbers with specified prime factors |
---|---|
Canonical name | DeterminationOfAbundantNumbersWithSpecifiedPrimeFactors |
Date of creation | 2013-03-22 16:47:41 |
Last modified on | 2013-03-22 16:47:41 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 12 |
Author | rspuzio (6075) |
Entry type | Theorem |
Classification | msc 11A05 |