Thursday, March 14, 2013

The Math Blog: Hexagonal Numbers

h_n= 2n^2-n = n(2n-1) = {{2n}\times{(2n-1)}\over 2}.\,\!

I will do the Math Blog every day, unless I am too busy. If I am too busy, then I will post twice on a day.

 Today is awesome math with hexagonal numbers! Hexagonal numbers are also pretty awesome, and this is the formula for deriving hexagonal numbers:

h_n= 2n^2-n = n(2n-1) = {{2n}\times{(2n-1)}\over 2}.\,\! (again, credit to Wikipedia)

All hexagonal numbers are triangular numbers, but only every other triangular number is a hexagonal numbers. Hexagonal numbers can be diagrammatically be represented at this link here.

From this, you can derive the first hexagonal numbers:

1, 6, 15, 28, 45...

Every even perfect number is hexagonal, as 6 and 28 are hexagonal and so is 496: you can solve this equation on your own using algebra. I will solve it because I feel like it. If you don't want to watch an algebraic proof, skip this section.

2n^2 - n - 496 = 0

I used a calculator at http://www.webgraphing.com/quadraticequation_factoring.jsp, FYI. The perfect number 496 is the 16th hexagonal number.

Try it yourself; 512 - 16 = 496.

In fact, no odd perfect number is found yet, so it comes to reasoning that all perfect numbers that we have found are hexagonal, and yet, that statement is CORRECT!

Bye!

~Albert Tam
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