Polynomial Expansion
I wrote an article up but don’t have the time to make sure my “graphics” don’t get messed up when I post it so you can find the document here: hart.gory.googlepages.com/MathUnderground.doc I am hoping for feed back.The Remainder Sequence, Part 1
Fri Aug 18, 07:11 PM by Noam Samuel
Well, as promised, I am here to write about the remainder sequence and its implications. In this article, I’ll pick up on the first question. Is the remainder sequence of a number unique per that number, or are there two numbers that hold the same remainder sequence?
Well, we can start off by saying that if I begin with any the number n, we know that if the remainder of n+m by division of x (also called n+m mod x) then m is divisible by x.
From then on, we can go to say that if we want to add a number x to another number n such that the remainders of x+n by division of all the prime numbers smaller than x will be equal to the remainders of x be division of all the primes smaller than x, n would have to be divisible by all primes smaller than x.
Now, since all primes are coprime (have a greatest common divisor of 1), we know that the smallest number divisible by all primes before x is their collective product. This is because of a little thing called the fundamental theorem of arithmetic (again, I’m going to let you figure out how that one works).
But wait! Suppose we take the product of all primes smaller than x and subtract 1 from it, we will now get a number that cannot be divisible by any prime smaller than x, and thus must be a prime in of itself, or divisible by two or more new primes that exist between the largest prime smaller than x and the multiple of all primes smaller than x.
Now, it doesn’t take much to understand that if p is the product of all primes smaller than x, then p By now you must ask “how does all this connect back to the Remainder Sequence”? Well, now comes your time. In case you have forgotten, the remainder sequence is the sequence of the remainders by division of x for all primes smaller than x for any number x. Now, since we now know that the smallest number that has the same remainder sequence as x does up to the remainder by division of the largest prime smaller than x is x+p, where p is the product of all primes smaller than x, and that there must be another prime between x and x+p (how do we know that it must be larger than x? The answer is surprisingly circular: we were discussing all the primes smaller than x, and thus the new prime cannot be smaller than x), we also know that, since there must be an additional element in x+p’s (and in all subsequent numbers’) remainder sequence, its remainder sequence (and that of all subsequent numbers) must be different. Thus, there cannot be any number larger than x which has the same remainder sequence. Since this applies to all numbers, this means that the remainder sequence must be unique per every number. QED. P.S.
You still haven’t answered my challenge: Can you construct a method to convert between two bases and prove that it works? I’m still waiting. Oh, and participation by writers in The Math Underground is prohibited.
Teaser: What The Remainder Sequence Holds In Store
Fri Aug 11, 02:28 PM by Noam Samuel
I swear that I was going to post a full article, but my family is going on a camping trip in a very short span of time. Thus, since I do not want to leave the Underground again without an article for a week, I’ve decided to post a little about an article soon to come.
You see, I was thinking about prime numbers for quite a while, and eventually I came to the conclusion that if we were to take the remainder left from divinding a number by each prime that comes before it, the number would be prime only if none of those were 0. While this might seem obvious and, quite frankly, useless, it can be quite interesting.
My next idea was to take any number n and construct a sequence of numbers representing the remainders from the division of n by each prime that is smaller than it, starting from 2 (which is, well, the smallest prime). This sequence was eventually called the remainder sequence of a number.
So, for example, the remainder sequence of 5 would be “1,2”. The remainder sequence of 15 would be “1,0,0,1,4,1”. The remainder sequence of 23 would be “1,2,3,2,1,10,6,4”.
At this point, most people wouldn’t get excited. This is, after all, just another representation of numbers. Nothing new here. But before you dismiss it as an idea deemed to the trashcan of mathematics, think of the following two question:
First of all, is it unique? We have no indication that only one number can hold each remainder sequence, but this is not to say that it isn’t true.
And, just as important, what would the implications be if it were unique?
I’ll explore both of those questions in my next article.
Prisoners, Games, and Kant
Sat Aug 5, 09:11 PM by Patrick N. R. Julius
Mathematics is, at its core, an extension of rational thought. It takes the basic principles of logic, and through definition and deduction alone, expands it into a complete system of reasoning in its own right.
But today I’d like to go back to the roots of mathematics, the question of what makes our choices rational.
The prisoner’s dilemma leaps to mind; it’s always an interesting puzzle to consider, and it forces us to question what rationality really is.
What’s this dilemma? It has many versions, but here’s mine—-it places the stakes a little higher than most. You have been captured by the Russian Mafia. You also learn that another person, Steve, whom you have never met, is likewise captured.
The Russians believe you and Steve are working together in a plan to blackmail one of their agents. You have no such plans, but they don’t believe you when you say so.
You are placed into an interrogation room, and asked to confess to collusion with Steve.
If you confess and Steve refuses, you will be released, but Steve will be executed. If you do not confess, and Steve does, you will be executed, and Steve will be released. If you both confess, you’ll both be tortured for the rest of your lives. But if you both refuse, you’ll both be detained for one year.
In other words, this is a bad situation, in which you have two bad options—-a dilemma.
Now, if we stick to mathematics as it is commonly understood, we go to game theory. In game theory, it’s every (hu)man for himself, and you’re out for your own best interest.
Well, consider the options Steve has:
If he confesses, you’ll be tortured one way, and executed the other way. So obviously the first way is better.
If he doesn’t confess, you’ll be set free one way, and detained the other way. Again, the first way is better.
Game theory’s answer is clear, for the Nash Equilibrium is obvious: you both confess. And hence you are both tortured. Nicely done.
Where does this go wrong? Wasn’t it rational to confess either way?
No, it wasn’t! It was totally irrational! You should keep your mouth shut!
Why? Because rationally runs deeper than mathematics. Life isn’t game theory—-and it’s not always about you.
Immanuel Kant understood this, and in formulating his theory of rational ethics, disregarded any notions of “self-interest” or “individual pleasure,” for he recognized something subtle but important: seeking your self-interest is not in your self-interest.
Sounds really weird, but think about it; if everyone in the world analyzed life with game theory, any time we got in a prisoner’s dilemma sort of situation, we’d all end up screwed.
As Kant realized, we can only truly achieve our best interest if everyone agrees to the rules. Society only works if the rules can be generalized, or as he said, “universalized.” You don’t act for your own self-interest in each and every decision; you act for your own self-interest in the formulation of general rules.
Hence, the general rule here would be, “do what’s best for both prisoners.” And so neither of you confess, and once the year is up, you’ll be let go, to go on with your life.
Now, what if Steve doesn’t feel this way? What if he decides to betray you, and confesses anyway? Ultimately, that’s out of your control. If Steve isn’t as Kantian as you, you might end up dead by his malfeasance. But at least you’ll have done the right thing.
And the only way society can work is by people doing the right thing, not the thing whis is “rational”—-which is only apparently so.
Besides, Steve’s conscience will catch up with him eventually.
Comment [1]
A Bit About Measuring Infinity, And The (Infinitely) Infinite Flop
Mon Jul 31, 06:53 PM by Noam Samuel
It’s hard to measure infinities. On one hand, it seems reasonable that the set of real numbers is larger than the set of natural numbers, but how would you define “larger infinity”?
One of the definitions of a set having more elements (or, as we mathies would say, a higher cardinality) than another is that there cannot exist a mapping of all the elements in the latter set into the former such that every element in the former set has an element in the latter set mapped to it, while there can be such a mapping from the former set to the latter.
For example, it is possible to say that the set of all real numbers is larger (or, more accurately, has a higher cardinality) than the set of natural numbers.
In fact, one of the categorizations of infinity is the differentiation between “countably” and “uncountably” infinite sets. Sets that are countably infinite must be mappable in a one-to-one mapping onto the set of natural numbers, whereas sets that are uncountably infinite must not be. The set of real numbers, for example is uncountably infinite, whereas the set of numbers divisible by 2 is countably infinite.
This, however, does not seem to give a full answer to the intriguing question of how one goes about measuring an infinity. Some infinities seem “larger” than others, but there is no known way to formulate that difference (oftentimes because it is provably not there). And how does one go about saying which set is more infinite than another, and how many infinities are equal to this or that? Well, this is where the second part of my title, the (infinitely) infinite flop comes in.
Despite its name, it is not all that horrible of a flop. The actual reason it is called the (infinitely) infinite flop, is because it is a flop about the subject of things that are (infinitely) infinite.
The idea came when I noticed that lots of infinite sets were infinitely as other infinite sets, meaning that per each elements in the latter set, there is an infinite amount of elements in the former. I came eventually to think that this property might be used to stratify infinite sets.
Thus, I formulated the following definition: Set P is infinite to set Q when there exists a function [f(x)] whose domain is Q and whose range is the set of the infinite subsets of P such that for every two elements a and b in Q, f(a) and f(b) are disjoint. For many practical purposes, however, the definition could just read as the following: “Set P is infinite to set Q if every element in Q maps to an infinite number of unique elements in set P.”
I even created a symbol for this relation: 
. In fact, one of the first complaints I got when I presented my idea to my friends was that the symbol was symmetric, and does thus not properly represent the asymmetry between P and Q. This, however, turned out not to be as much of a problem as we thought.
You see, it starts from the fact that such a stratifying (or, at the very least, positioning) relation cannot, in any way, be reflexive. If it turned out that 
for any P, the whole purpose of constructing the definition goes down the drain.
It didn’t take me long to realize this, and I was soon off to try to prove and disprove the existence of such a “reflexive P”. The task, however, was not easy, and after a few weeks of not being able to do either, I abandoned the idea altogether.
But some time after the creation of The Math Underground, Patrick and I were pining for ideas about which we could write, and I remembered my definition of infiniteness. Without the missing proof, however, I could publish the article just to find out later that the whole idea was useless.
Eventually, we decided to leave it as an open question for the reader, and then went into the nuances of the symbols I used and other parts of the definition when suddenly it hit me.
First of all, there’s an infinite number of prime numbers (there’s proof of that, but I’m going to “chicken out” again since I’m too lazy to write it), each of which has an infinite number of positive integer powers. Now, the set of the positive integer powers of one prime must be disjoint set of positive integer powers of any other prime due to a little thing called the fundamental theorem of arithmetic (I’ll let you people figure out how that works).
Now, if I were to create a function, f(x), such that for every natural number n, f(n) would be the set of all positive integer powers of the nth prime, it would obey all the rules I laid out in my definition: Each natural number is mapped to an infinite number set of natural numbers, and per every two numbers, the sets mapped to both numbers are disjoint. Thus, 
.
But there’s more: remember the countably and uncountably infinite sets? Well, through a simple isomorphism you could extend that to all countably infinite sets, thus rendering the whole thing even more ridiculous (and, essentially, since there is an infinite number of countably infinite sets, an infinite number of flops, each of which infinite in its own right).
If I were a Hollywood supervillian, I would have but one thing to say: “My plans are ruined! Nooooo!”