Responses to Blog Prompts #1 and #2

Okay everyone, so this is going to contain a few brief comments concerning the first two prompts.

First things first, most everyone did fine with their responses (as like I mentioned, I was only asking for your points of view for these two. Do not expect these to stay QUITE as open ended as the year progresses.) I would like to note that while short answers ARE acceptable, I prefer to see the brief explanation to full paragraph length answers. This really isn't a lot of work to do once a week (it should take you no more than 20 minutes at this point) and I've been working on one weekend's worth of homework for a total of about 18 hours over the past two days, so I don't want to hear too many complaints about that. ;p

Okay, so moving on:

What is Number Theory exactly? A lot of you hit the nail on the... side of the head. What I mean is that you got the gist of it while missing some major points. Number theory IS the study of numbers and their behaviors, but it deals with a lot more than just sieves, lcms, gcds, prime numbers, and "harder" arithmetic. Number theory essentially connects everything together. You'll certainly be getting this feeling later on when dealing with things like the Chinese Remainder Theorem which is essentially a shortcut to solve systems of congruences when dealing with modular arithmetic (which should already clearly illustrate that this is a lot more than simple computation.) Furthermore, and especially recently, number theory has been applied in an ever-growing number of fields. Cryptosystems have long been a clever application of number theory and its principles, but more recently ideas have been applied to nearly all fields of applied mathematics (including physics, biology, chemistry, and even things like quantum computing and whatnot.) In a nutshell, there is a LOT you can do with number theory and the basic understanding of numbers doesn't hurt too much either.

For prompt number 2,
everyone was able to calculate the first 10 terms of the sequence (hopefully that is, I know some people who are far too quick to reach for their trusty ctrl+c/ctrl+v combo.) Unfortunately, it seems that no one was able to get the exact pattern without having seen it before. Certainly each number is the previous sum plus the current index (n,) but that's a tad too obvious. The closed form representation of the series on the other hand is a bit more elusive. Malerie, you came close by noting that n^2-n was 6 in the case where n=3. If you had gone a bit further with this you would've noticed that n^2+n was 12 in this case which happens to be twice the current sum. Furthermore, we can note that n^2+n is ALWAYS 2a_n (where a_n is the current sum.) A couple of seconds should allow everyone to convince themselves that this means that

(n^2+n)/2=a_n

That being said, if I want to know the sum of the first 1000000 positive integers, we simply plug in 1000000 and get
(1000000^2+1000000)/2
=(1000000000000+1000000)/2
=(1000001000000)/2
=500000500000

Now, how do we prove such a thing? There are a ton of ways, but two main ones that I'll mention. One is intuitive, the other is more mathematical. In the future if I ask you guys to prove (or try to prove) anything, you can go down either of these routes. Intuition can take you a long way.

Think of adding numbers like adding dots. We start with one dot, then add two, then three, and so on.

Let's consider the case that we have n=10, we can construct this picture:

.
..
...
....
.....
......
.......
........
.........
..........

Clearly this pattern will continue on if we continue adding more dots. What we get is nothing but a triangle. Now let's try and find something we can calculate the area of easily. What's the easiest in this case? Squares/rectangles come to mind, so let's turn this into a square with red dots representing things we've just added.

...........
...........
...........
...........
...........
...........
...........
...........
...........
...........

As you can see, all this is is a mirror image of the number of dots we want to find. But what we have formed is a rectangle with 10 rows and 11 columns. What is the area of such a rectangle? Hopefully you're thinking to yourself "length*width" and if so, you're correct. So what is the area of the black part of the rectangle? Exactly half of that, right? So all we need is
length*width/2

In this case, the length is 10 and the width is 11 so there are 10*11/2=55 total black dots.

It shouldn't take long to convince yourself that this will hold for any number of rows. The length will always be n and the width will be n+1.

This means that our formula is a_n=n(n+1)/2

As for a more mathematical proof, let's write out what we know.

a_n=1+2+3+4+...+(n-2)+(n-1)+n
and if we change the order around
a_n=n+(n-1)+(n-2)+...+4+3+2+1
So we can add these two equations and get
2a_n=(n+1)+(n+1)+(n+1)+...+(n+1)
But clearly we only have n terms (so we have n of these (n+1)'s)
so that means we have 2a_n=n*(n+1)
or in other words:
a_n=n*(n+1)/2


So there are two different proofs. I should note that a lot of you mentioned you would just plug in a bunch of numbers and test it. This is NOT a reliable method of proof. I could find you infinitely many more formulas that "almost" work for this specific pattern but at some single point fail. Also, a lot of you mentioned that there must be a formula because there is a pattern. Ah, that would be nice wouldn't it? Unfortunately this is NOT always the case.


Consider the following sequence:
3,1,4,1,5,9,2,6,5,3,5,8,9,7,9,3,2,3,8,4,6,2,6,...
There is most definitely a pattern here, each term is the next digit in pi, or in other words that it is the next decimal place in the circumference of a circle divided by its diameter. Unfortunately, there is no algebraic formula for this sequence that does NOT involve the usage of pi itself. If anyone can come up with one I'd be happy to hear it (and claim whatever prize comes for breaking all of mathematics.) That said, do not assume that just because there is a pattern there must be a formula to it. A few people mentioned that "no one would want to sit and add all of those numbers." I can assure you that no one wants to calculate the one trillionth digit of pi by brute force and would LOVE to just plug in their number into a formula, but sometimes things just have to be done the hard way.

That's all for now, I'll be thinking of another question for next week. See you all then!

Ciao!