Perfect, Triangular, and Hexagonal NumbersDate: 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/ |
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