Triangular Day: Some More Fun Facts

Today is another triangular day. It is September 15 and 15 is a triangular number. I'd like to continue from two weeks ago and show a few more fun facts.

First off, remember the perfect numbers that we talked about last week? Those are the numbers whose factors add up to itself, like 6, 28, and 496. Let's see if any of them are triangular. We'll see if they fit the formula we learned a couple months ago: n(n+1)/2.

6 = (3)(4)/2
28 = (7)(8)/2
496 = (31)(32)/2
8128 = (127)(128)/2

Clearly, all of these numbers are triangular. Turns out, every perfect number is a triangular number.

This one actually isn't too hard to prove. Remember last week when we were generating perfect numbers? We would multiply a power of two by one less than the next power of two. For 6, we did 2 x 3. For 28, we did 4 x 7.

Well, let's say that the seven is n. Then, what would the (n + 1)/2 equal?

(7 + 1)/2 = 4

So, we are doing 4 x 7, which is what we were doing before. What about for the 2 x 3?

(3 + 1)/2 = 2

How about for 496?

(31 + 1)/2 = 16

Basically, with the method we used for generating perfect numbers, we are following the triangular number formula; n(n+1)/2.

Let's look at another pattern. Soon, we will be learning about other number systems, like the binary system and the hexadecimal system. Basically, some numbers are written differently in other systems. We will take some numbers with all ones and convert them from base nine to base ten.

1 = 1
11 = 10
111 = 91
1111 = 820
11111 = 7381

This will make sense in a few months. Anyways, let's see if these numbers are triangular.

1 = (1)(2)/2
10 = (4)(5)/2
91 = (13)(14)/2
820 = (40)(41)/2
7381 = (121)(122)/2

These are all triangular as well. This one, I don't have a proof for, but if you know one, please let me know.

One more quick fact: let's look for some prime triangular numbers. Here is the sequence:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153,...

Do you find any primes? Well, three. Anything else?

Turns out, three is the only prime triangular number. Again, I don't have a proof for that one, but I still thought it was pretty cool.