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As mentioned above, relationships between the two subjects of the title are emphasized.

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The book is nicely broken up into manageable sections that would fit well into a lecture course. Interdependences among chapters are clearly indicated. The topics are presented clearly and logically with relationships among them clearly pointed out and discussed in detail. All pages display very well on my screen, with no legibility or distortion issues that I could see. This is not relevant for a mathematics text, but I saw nothing that would be offensive to a reader of any ethnic background.

The text is so comprehensive that it feels overwhelming. The author wanted to include all of the mathematics required beyond a standard calculus sequence. However, the mathematical maturity required to read and learn from this text is quite However, the mathematical maturity required to read and learn from this text is quite high. The first two chapters cover much of a standard undergraduate course in number theory, built up from scratch. However, it almost completely lacks numerical examples and computational practice for the students, which would give those new to the material time and experience in which to digest, assimilate, and understand the material.

I would think that a book targeted at this level of mathematical sophistication would assume students are comfortable with for example the most basic notions of group theory or the idea of equivalence classes. I can't imagine an appropriate audience for this text: one with the ability to read and work entirely at this abstract level but without any or most of the mathematical preparation provided in at least half the chapters. I found no mathematical errors. The mathematical presentation is rigorous, clear, and well-explained.

MATH1: Introduction to Number Theory | Academic Calendar

It can be terse at times, skipping steps and making conceptual leaps that will be challenging for all but the very best students. The book covers both standard background that will always be relevant for these topics: the number theory and algebra background, the probability theory. The computational chapters use pseudocode, so they will not be quickly outdated when new languages become fashionable.

Most of the algorithms studied are quite "classical" as much as that makes sense for computer science , with modern ideas and developments usually relegated to "Notes" at the end of the computational chapters.

Math 406 - Introduction to Number Theory

This will, of course, become outdated with new research in computer science. But any faculty member who keeps up with the relevant research will be able to mention new developments to students, and it will not interrupt the flow of the ideas at all. The book is exceedingly well written, though it is at a very high level.

It is not "friendly" or "chatty" as you will find with many number theory books targeted to undergraduates. For many students this will detract from clarity because they do not yet have the mathematical sophistication to work at this level. The book does an excellent job of consistency of notation. For example, it starts with a development of number theory concepts, and develops notation for residue classes in the integers modulo n.

Later in the chapters on groups and rings, this same notation is used in more general situations. Whenever there is the potential for confusion for example, in using "a mod b" as a binary operation as is common in computer science versus using "a is congruent to x mod b" as is more standard in mathematics the author is careful to point out the dual meanings and to warn the reader that there is some overloading of terminology.

It is unavoidable that this will happen in any book that treats both subjects seriously, and the author is careful with notation and keeps potential confusion to a minimum. The book has 21 chapters, each with several sections. Most, but not all, sections end with a set of exercises. Essential exercises are underlined a very nice feature! What would be helpful would be some suggested paths through the text for various purposes.

Introduction to Number Theory

I don't think it would be appropriate in any class to start at Chapter 1 and and work through all or even most of the content. I imagine that most classes would skip the background material and head straight for the computational chapters, with the background there "as needed" for the students. My main comment about the structure is that the mathematics chapters and the computational chapters seem to be separated. For example, the chapter on "Congruences" covers a tremendous amount of number theory, not all of which falls naturally in my mind under that heading.

Chapter 1 has a section on "Ideals and greatest common divisors," but Euclid's Algorithm is not tackled until Chapter 4 a more computational chapter.

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As I read, I often felt "now we are doing mathematics I read a standard PDF file. There were a few hyperlinks from the table of contents to section headings, for example , but not much else in the way of interface. Everything was rendered clearly. There is a lot of interesting history and "cultural" notes in the computing chapters, and almost none in the more mathematical chapters. A student who studied from this text would miss a lot of the standard "mathematics culture" communicated in a more traditional number theory course.

My primary comment is that I cannot pin down the audience for this book. I could not use this in an undergraduate number theory class; it is at far too high a level and moves far too quickly. I could not use it in a graduate number theory class; it assumes no background at all and does not do some standard topics. I suppose it would be useful for self-study by a very advanced student who already knew a good deal of mathematics and wanted to explore the computational side. We were able to prove that any amount greater than 27 cents could be made. You might wonder what would happen if we changed the denomination of the stamps.

What if we instead had 4- and 9-cent stamps? Would there be some amount after which all amounts would be possible?

Well, again, we could replace two 4-cent stamps with a 9-cent stamp, or three 9-cent stamps with seven 4-cent stamps. In each case we can create one more cent of postage. Using this as the inductive case would allow us to prove that any amount of postage greater than 23 cents can be made. What if we had 2-cent and 4-cent stamps. Here it looks less promising. If we take some number of 2-cent stamps and some number of 4-cent stamps, what can we say about the total? Could it ever be odd? Doesn't look like it. Why does 5 and 8 work, 4 and 9 work, but 2 and 4 not work?

What is it about these numbers?

If You're an Educator

If I gave you a pair of numbers, could you tell me right away if they would work or not? We will answer these questions, and more, after first investigating some simpler properties of numbers themselves. It is easy to add and multiply natural numbers. If we extend our focus to all integers, then subtraction is also easy we need the negative numbers so we can subtract any number from any other number, even larger from smaller. Division is the first operation that presents a challenge.

If we wanted to extend our set of numbers so any division would be possible maybe excluding division by 0 we would need to look at the rational numbers the set of all numbers which can be written as fractions. This would be going too far, so we will refuse this option. In fact, it is a good thing that not every number can be divided by other numbers. This helps us understand the structure of the natural numbers and opens the door to many interesting questions and applications. It is either true or false. Negative numbers work just fine for the divisibility relation. While we don't really know how to divide, we do know how to multiply.

Is this the best we can do? How far are we from our desired ? In other words, we have found that. It turns out that the process we went through above can be repeated for any pair of numbers. We know this because we know about division with remainder from elementary school. This is just a way of saying it using multiplication. Due to the procedural nature that can be used to find the remainder, this fact is usually called the division algorithm :. We get the infinite set. Well, certainly 1, does, as does 6, and Thus we get the remainder class.

There are three more to go. Note that in the example above, every integer is in exactly one remainder class. All fun technical language aside, the idea is really simple. If two numbers belong to the same remainder class, then in some way, they are the same. With all this in mind, let's introduce some notation. It works if we are thinking division by 5, so we need to denote that somehow. What we will actually write is this:. Of course then we could observe that. Turns out, it doesn't matter: they are equivalent. So the remainders on the left-hand side must cancel out.

That is, the remainders must be the same. It will also be useful to switch back and forth between congruences and regular equations. The above fact helps with this. You should take a minute to convince yourself that each of the properties above actually hold of congruence. Try explaining each using both the remainder and divisibility definitions. Next, consider how congruence behaves when doing basic arithmetic. What if we add something congruent to 1 to something congruent to 2? Will we get something congruent to 3? The above facts might be written a little strangely, but the idea is simple.

If we have a true congruence, and we add the same thing to both sides, the result is still a true congruence. This sounds like we are saying:. One of the important consequences of these facts about congruences, is that we can basically replace any number in a congruence with any other number it is congruent to. Here are some examples to see how and why that works:. We could do long division, but there is another way. Why is this okay?

So we can in fact replace the with simply a 4. The above example should convince you that the well known divisibility test for 9 is true: the sum of the digits of a number is divisible by 9 if and only if the original number is divisible by 9. In fact, we now know something more: any number is congruent to the sum of its digits, modulo 9.

Thus we can simplify further:. So far we have seen how to add, subtract and multiply with congruences. What about division? There is a reason we have waited to discuss it. It turns out that we cannot simply divide. Now that we have some algebraic rules to govern congruence relations, we can attempt to solve for an unknown in a congruence. Let's also see how you could solve this using our rules for the algebra of congruences. Such an approach would be much simpler than the trial and error tactic if the modulus was larger.

First, we know we can subtract 2 from both sides:. Then to divide both sides by 3, we first add 0 to both sides. This gives,. Now divide both sides by 3. All we need to do here is divide both sides by 7.

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We add 13 to the right-hand side repeatedly until we get a multiple of 7 adding 13 is the same as adding 0, so this is legal. So we have:. Here, since we have numbers larger than the modulus, we can reduce them prior to applying any algebra. Now divide both sides by 9. So the solutions are those values which are congruent to 8, or equivalently 3, modulo 5.

This means that in some sense there are 3 solutions modulo 3, 8, and We can write the solution:. We could now divide both sides by 3, or try to increase 9 by a multiple of 14 to get a multiple of 6.