Perfect, Triangular, and Hexagonal Numbers

Date: 12/18/2001 at 16:57:55
From: Rachel
Subject: Perfect numbers

How are perfect numbers related to triangular numbers and hexagonal 
numbers?

Date: 12/19/2001 at 03:04:55
From: Doctor Floor
Subject: Re: Perfect numbers

Hi, Rachel,

Thanks for writing.

In the following answer from the Dr. Math archive, you can find a 
formula for even perfect numbers:

   Perfect Number
   http://mathforum.org/dr.math/problems/tadeu.7.21.96.html   

The formula is

   N = (2^(n-1))(2^n - 1)

provided that 2^n - 1 is a prime. Now, when we let X = 2^n - 1, then 
X+1 = 2^n and thus N = X(X+1)/2. That is exactly the formula for a 
triangular number. For that formula, see:

   Formula for Triangular Numbers
   http://mathforum.org/dr.math/problems/huff8.29.98.html   

We conclude that a perfect number is always a triangular number.

A more general formula for polygonal numbers is given at:

   Figurate and polygonal numbers
   http://mathforum.org/dr.math/problems/megan11.21.98.html   

From this we can easily find the formula for hexagonal numbers:

  H = 2r^2 - r = r(2r-1)

Taking r = 2^(n-1) we see that 2r-1 = 2^n - 1 and we easily conclude 
that a perfect number is always a hexagonal number as well.

So each perfect number is a hexagonal as well as a triangular number.

If you need more help, just write back.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/