Product-Perfect Numbers

Date: 02/26/2001 at 14:13:51
From: Mike Smith
Subject: Product perfect numbers

This is the question: Is there a statement to describe product perfect 
numbers? 

We call a number a product perfect number if the product of all its 
divisors, other than itself, is equal to the number. For example, 10 
and 21 are product perfect numbers since 1*2*5 = 10 and 1*3*7 = 21, 
whereas 25 is not, since the product of its divisors, 1*5 = 5 is too 
small. I have listed all the product perfect numbers between 2-60, 
which are: 

6,8,10,14,15,21,22,26,27,28,33,34,35,38,39,44,45,46,51,52,55,57,58. 

I am stumped on the description - a description that should make it 
easy for you to produce a six-digit product perfect number with very 
little work. 

Thank you, 
Mike

Date: 02/26/2001 at 20:15:43
From: Doctor Schwa
Subject: Re: Product-perfect numbers

Hi Mike,

You've made a lot of good progress.

As a hint, I'll suggest that you look at the prime factorization of
these numbers, and compare them to the prime factorizations of the
numbers that aren't product-perfect. Perhaps by the time you get up
to 20 or so you'll see a pattern... if not, please write us back
for another hint.

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/   

Date: 02/26/2001 at 23:31:29
From: Mike Smith
Subject: Re: Product-perfect numbers

I did all the prime factorizations of the numbers 2-20 and found that 
all of them have two prime factors except for 8, which is 4*2*1. 

I don't understand. I still can't find a pattern because 27 is 3*9 and 
9 isn't prime. Can you give me another hint? 

Thank you, 
Mike

Date: 02/27/2001 at 12:31:31
From: Doctor Schwa
Subject: Re: Product-perfect numbers

Great work! You discovered correctly that a number is perfect if, and 
only if, it is one of two types: either it is two different primes 
multiplied together, or it's a number like 8 or 27. What do those 
numbers have in common?

Take a look at their prime factorizations...

Enjoy,
- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/   

Date: 02/27/2001 at 17:09:58
From: Mike Smith
Subject: Re: Product-perfect numbers

Is it that the prime factorizations are a prime and its squared 
number multiplied?

Thank you, 
Mike

Date: 02/27/2001 at 20:25:32
From: Doctor Schwa
Subject: Re: Product-perfect numbers

Exactly.

A shorter way to say the same thing might be: a number is product 
perfect if and only if it is: either the product of two different 
primes, or the cube of a prime.

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/