Exploring Number Theory: Study of Integers

Number Theory 1 Number Theory is the branch of mathematics concerned with the study of the integers. Number theorists are interested in properties, relationships, and generalizations involving integers. The development of this field has been closely intertwined with the historical development of whole numbers, integers and rational numbers. In many ways, number theory can be viewed as a branch of pure mathematics; however, more recently it has become a part of applied mathematics. In fact, number theory is now often defined as a branch of applied mathematics that specializes in problems concerning the integers. In addition to its own history, number theory is also considered to be connected to many other areas within mathematics such as algebraic number theory, arithmetic algebraic geometry (AARH) and diophantine approximation. In mathematics, the integers and the real numbers are convenient domains from which to illustrate proofs.Those fields are number theory and elementary real analysis. The principal advantage is that example domains are available to use. Let us begin by considering the integers. The development of the mathematical study of the integers, a branch of mathematics known as number theory that is concerned with properties exhibited by integers other than calculation, began around 700 BCE. The study has contributed to one of the most significant branches of Pure Mathematics. Number Theory. Number theory is filled with problems that are easy to state but difficult to solve, and many remain unsolved. However, some of the results have applications in modern life. Internet security being perhaps the most important. In mathematics, the interest lies not in their use as a counting device, but in the arithmetical system they create. For any two integers (positive or negative whole numbers), the result of adding them together, subtracting one from the other, or multiplying them will always be another integer. Division, however, is more complex. This is where things get particularly intriguing. If a pair of integers is divisible by 5, and the second integer is divisible by 15, then the first integer is divisible by 5. The integer 3 is the result of dividing 15 by 5. When dividing two integers, such as 7 and 15, you can only get whole-number results if you are willing to allow fractional results. When division is restricted to the integers, it actually relates to two numbers, a quotient and a remainder. In mathematics, when you divide nine by four, you get a quotient of two and a remainder of one. The number nine can be expressed as the product of four, two, and one—a special case of the first theorem concerning integers. The division theorem. The division theorem states that if a and b are integers with b greater than 0, then there exist unique integers q and r such that a = qb + r and 0 less than or equal to r less than b. The theorem has two parts. There is an existence proof for sets of integers which exhibit this property. And there is a uniqueness property associated with integer pairs, since these pairs are unique in satisfying the given property. In order to prove the theorem, we'll take each step one at a time. We will first prove the existence of the function, and then we will prove that it is unique. It is always a good idea to begin a proof by stating the method you will use. Let all nonnegative entities of the form a-kb be considered as integers. Make sure that at least one of them is less than b. The theorem states that among the integers a-qb, there exists one with r between 0 and b. Let us consider all possible candidates for a minus k b or a minus q b, and prove that among those candidates one satisfies the given condition. The k for which the equation is true will be the q that we choose to take with the k. One must first show that there are such integers. For example, consider the equation k equals negative absolute value of k. Next, since b is greater than or equal to 1, a - kb = a + |b| = a + b, which is greater than or equal to a + |a|, which in turn is greater than or equal to 0. By finding the integer k that gives us a number with this property, we can choose the smallest number of this kind. Let r be the smallest integer satisfying the equation and let q be the value of k for which it holds true. Under these conditions, we have r = a - qb. With this relationship defined, we can indeed say that a is equal to qb plus r. If it can be shown that r lies between zero and b, then it will have been proven that the function exists. We complete the proof by showing that r < b. But if we assume that r is greater than or equal to b, in other words we use proof by contradiction and show that r is less than b. If r is greater than or equal to b, then a – q + 1 b = a – qb – b, which equals r – b, which is greater than or equal to 0 by the assumption stated above. Thus, the value of a- (q + 1)b is a non-negative integer of the form a-(k-1)b. Thus r is the smallest such number and a-(q+1)b < a-qb, which is r. This is a contradiction. If r is as small as this, we cannot find a value of r smaller than it. Since there is a contradiction, r cannot be equal to b. This proves that r is less than b. If you are unfamiliar with the proof, then you may find the argument difficult to follow. It is not hard to understand; it just has a complicated structure. The goal is to look at all of the possible candidates to make this statement true. First, we need to demonstrate that a candidate exists. By exhibiting one, we do just that. In order to satisfy this additional requirement, we pick the one where r is the smallest; it is the minimum value of r which gives us one of these things. Next, we show that it satisfies this requirement by showing that it is minimal. Elementary algebra is just a step in the argument; it's necessary to take each step before moving on to the next. There are many arguments in mathematics like this one. They involve only basic arithmetic and algebra, but the structure of the argument is intricate enough to prove its point. Okay, well that proves that something exists. Now we must determine if it is unique or not. To prove that a number is unique, we show that if there are two representations of the number—for example, a = qb + r, where 0 < r < b and r is prime and less than b—then r = r prime, and q = q prime. The first part of the division theorem shows that such representations exist. We are now showing that if there are two representations of a function, then they are actually the same. The r is the same and the q is the same. The letters q and r are used to denote quotient and remainder, respectively. We have not yet defined quotients and remainder. Dividing one integer by another. Let's review the equation we have here. Qb + r = q prime b + r prime. Rearranging the equation, we get r prime - r = b(q - q prime). Let us label that equation 1 for later reference. Now we take the absolute value of both sides of this equation; when we do that, we get r prime minus r equals b times |q – q prime|. Let us label that equation 2 for later reference. If you remember that b is positive, then when you check absolute values you will find that b remains the same. So -b will be less than -r which will be less than or equal to 0, which is less than or equal to r prime, which is less than b. Thus, b is less than r prime minus r, which is less than b. If r prime minus r is between negative b and plus b, then the absolute value of r prime minus r is less than or equal to b. Thus, by 2b times the absolute value of q minus q prime is less than b. We can divide the absolute value of q-q prime by b to give us, q-q prime is less than 1. But all of the values here are integers, and in order to have a pair of integers with an absolute value difference less than one, their absolute value must be zero. This means that if q equals q i, then r equals r i , and equation one proves the uniqueness of r. And we have proven the division theorem. The focus of this section is the proof that the division property holds for all pairs of integers. The skill of mathematical proof is developed through the practice of proving simple results. Once mathematicians gain confidence in the method of proof, they are able to accept results that are not at all obvious. One example of an unexpected result was in the late 19th century, when mathematician David Hilbert described a hypothetical hotel with an unusual property. Hilbert's Hotel, a hypothetical hotel with an infinite number of rooms, has come to be known as the ultimate hotel. Like most hotels, the rooms at this hotel are numbered using natural numbers—one, two, three and so on. All the hotel's rooms are occupied on one particular evening when an unexpected guest arrives. "I'm sorry," says the receptionist, "but all our rooms are filled." You will have to travel elsewhere. The mathematician thinks for a moment before offering a solution. He says, "There is a way for you to give me a room without having to reject any of your existing guests." The clerk is reluctant to make such a change, but asks the mathematician to explain how it might be done. The mathematician proposes a solution. Each guest moves into the room number that is one more than his or her current room number, so the guest in room 1 moves into room 2, the guest in room 2 moves into room 3, and so on through the hotel. The occupants of room n usually moved into room n + 1. When that was full, room 1 became vacant. You put me in that room. The clerk considers the problem for a moment, then acknowledges the solution. A hotel can accommodate an additional guest in a completely full hotel without rejecting any guests if the new person shares a room with someone who is already staying there. With his logic, the mathematician persuades the innkeeper to give him a room for the night. The Hilbert hotel argument rests on the idea that a hotel with infinitely many rooms can exist. In fact, in order to illustrate one of the properties of infinity, Hilbert devised the following story. The importance of understanding infinity is that it allows one to deduce calculus, which is the foundation of modern science and engineering. After you've understood Hilbert's solution, consider the following variants. In the first variant of the Hilbert's Hotel scenario, all the rooms in the hotel were full when two additional guests arrived. How can they be accommodated in separate rooms, without anyone being asked to leave? A second variant: The clerk is again faced with a challenge. The hotel has no rooms available, but an infinite tour group arrives, each member wearing a badge that says, "Hello, I am n," for each natural number n equals 1, 2, 3, etc. Given that there is no room for any guest besides these infinite members of the tour group and that it would be unacceptable to eject any guest from their room even temporarily, can the clerk find a way to give all of these new guests a room to themselves without having to eject any existing guests? In the example of Hilbert's Hotel, we can see the importance of strict proof. Verification is a mathematical tool used to check the validity of an already known result, such as the division theorem. When used to verify obvious results, it may seem frivolous. When applied to an issue we are not familiar with, however—such as questions involving infinity—a rigorous proof is the only thing we can rely on. Now, we shall return to the division theorem. The division theorem in this form applies only to division by a positive integer b. So, the general division theorem lets us assume that a and b are integers with b ≠ 0. In particular, there exist unique integers q and r, such that a = qb + r and 0 < r < |b|. The general division theorem, like the previous division theorem we proved, is based on the principle of divisibility. We require only that b be non-zero, not strictly positive. Next, we will find the absolute value of b. The proof is fairly straightforward from the previous result. Thus, if b is an integer greater than 0, then the previous theorem shows that there are unique integers q and r such that b = qr. The equation a=q prime times the absolute value of b + r prime and 0 less than or equal to r prime, less than absolute value of b is possible. Now we can make the substitutions q = -q prime, and r = r prime. The absolute value of b, in this case a negative b, is the difference between qb and r. The result is that a is equal to qb + r, where 0 is less than or equal to r. This theorem simply follows from the previous results. According to the General Division Theorem, we can officially give names to the two numbers q and r. We will call q the quotient of a divided by b. And r is called the remainder. Thus we've come full circle, returning to the concept taught in elementary school about dividing numbers and the existence of quotients, remainders and so on.