## Saturday, September 1, 2012

### Triangular Day: A Few Fun Facts

Today is a triangular day. It is September 1 and 1 is the first triangular number. Rather than going through a whole proof, I thought that I would just go through a few fun facts.

First off, let's look at the sequence ignoring 1. We have:

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

Let's try to find some square numbers. We can find 36. If you keep going ahead, you find:

1225
41616
1413721
48024900
1631432881

Clearly, there are several squares that are also triangular. What about cubes?

Can you find any? No? There are actually no cube triangular numbers other than one.

What about fourth power, or fifth power, or sixth power? Turns out that there are absolutely none.

Here's another fun fact. Triangular numbers never end in 2, 4, 7, or 9. I'm not sure why, but they never do.

Finally, if you remember back to the divisibility post, we added up all of the digits in a number frequently to find out if it's divisible by any multiples of three. We can also use it for modular arithmetic, which I will probably talk about at some point in the near future. The adding of all of the digits in the number is commonly referred to as digital roots.

Let's find the digital roots of the triangular numbers.

1 = 1
3 = 3
6 = 6
10 = 1 + 0 = 1
15 = 1 + 5 = 6
21 = 2 + 1 = 3
28 = 2 + 8 = 10 = 1 + 0 = 1
36 = 3 + 6 = 9
45 = 4 + 5 = 9
55 = 5 + 5 = 10 = 1 + 0 = 1
66 = 6 + 6 = 12 = 1 + 2 = 3
78 = 7 + 8 = 15 = 1 + 5 = 6
91 = 9 + 1 = 10 = 1 + 0 = 1

What digital roots have we found. There are ones, threes, sixes, nines, and that's it. In fact, those are the only digital roots of triangular numbers. Again, I'm not sure why, but if you know a proof for this or any of the above facts, please comment it. But even without a proof, these facts are pretty cool.

1. these are surely funny facts lol :P
cool math 4 kids

2. Proof for the last digits of triangle numbers:

Every triangle number is of the form n*(n+1)/2 for some integer n.

If n = m, then n*(n+1)/2 ~= m*(m+1)/2 (mod 5) (where "~=" means congruent).

0*1/2 ~= 0 (mod 5)
1*2/2 ~= 1 (mod 5)
2*3/2 ~= 3 (mod 5)
3*4/2 ~= 1 (mod 5)
4*0/2 ~= 0 (mod 5)

Therefore, for every integer n, n*(n+1)/2 is congruent to one of {0,1,3} (mod 5). Hence n*(n+1)/2 is always congruent to one of {0,1,3,5,6,8} (mod 10), so the last digit is one of those numbers. 2, 4, 7, and 9 are impossible, as you observed.

3. Proof for digital roots of triangle numbers:

The digital root of a number is its remainder mod 9, with the exception that if a number is a multiple of 9 then its digital root is 9, not 0 (well known, look it up for a proof).

If n = m, then n*(n+1)/2 ~= m*(m+1)/2 (mod 9).

0*1/2 ~= 0 (mod 9)
1*2/2 ~= 1 (mod 9)
2*3/2 ~= 3 (mod 9)
3*4/2 ~= 6 (mod 9)
4*5/2 ~= 1 (mod 9)
5*6/2 ~= 6 (mod 9)
6*7/2 ~= 3 (mod 9)
7*8/2 ~= 1 (mod 9)
8*0/2 ~= 0 (mod 9)

Therefore the possible remainders mod 9 of triangle numbers are {0,1,3,6}, so the possible digital roots of triangle numbers are {9,1,3,6}.