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An Adventurer's Guide to Number Theory

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Overview

In this delightful guide, a noted mathematician and teacher offers a witty, historically oriented introduction to number theory, dealing with properties of numbers and with numbers as abstract concepts. Written for readers with an understanding of arithmetic and beginning algebra, the book presents the classical discoveries of number theory, including the work of Pythagoras, Euclid, Diophantus, Fermat, Euler, Lagrange and Gauss.
Unlike many authors, however, Mr. Friedberg ...

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Overview

In this delightful guide, a noted mathematician and teacher offers a witty, historically oriented introduction to number theory, dealing with properties of numbers and with numbers as abstract concepts. Written for readers with an understanding of arithmetic and beginning algebra, the book presents the classical discoveries of number theory, including the work of Pythagoras, Euclid, Diophantus, Fermat, Euler, Lagrange and Gauss.
Unlike many authors, however, Mr. Friedberg encourages students to think about the imaginative, playful qualities of numbers as they consider such subjects as primes and divisibility, quadratic forms and residue arithmetic and quadratic reciprocity and related theorems. Moreover, the author has included a number of unusual features to challenge and stimulate students: some of the original problems in Diophantus' Arithmetica, proofs of Fermat's Last Theorem for the exponents 3and 4, and two proofs of Wilson's Theorem.
Readers with a mathematical bent will enjoy and benefit from these entertaining and thought-provoking adventures in the fascinating realm of number theory. Mr. Friedberg is currently Professor of Physics at Barnard College, where he is Chairman of the Department of Physics and Astronomy.

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Product Details

  • ISBN-13: 9780486281339
  • Publisher: Dover Publications
  • Publication date: 1/9/1995
  • Edition description: Reprint
  • Pages: 228
  • Sales rank: 464,261
  • Product dimensions: 5.42 (w) x 8.45 (h) x 0.49 (d)

Read an Excerpt

AN ADVENTURER'S GUIDE TO NUMBER THEORY


By RICHARD FRIEDBERG

Dover Publications, Inc.

Copyright © 1994 Richard Friedberg
All rights reserved.
ISBN: 978-0-486-15269-1



CHAPTER 1

Seven jogged my elbow


Oil, Cake, and Glass

Every number has its character. I picture 7 as dark and full of liquid, like oil when it oozes from the ground. Three is lumpy and hard, but dark also, and 4 is soft and doughy and pale. Five is pale but round like a ball, and 6 is like 4 only richer, like cake instead of dough.

These are only my ideas. Perhaps you see 7 as a pincushion, and 5 as a bright spot of light. Everyone has his own pictures. Here are some more of mine.

Two is solid and tingly, like the Liberty Bell. Twelve is like a more delicate bell, of glass instead of iron. Eight is rough and hard like a stone, and 10 is smooth like a pebble on the beach. Nine is tingly but much sharper than 2 or 12, and it seems ready not only to ring but to shatter and burst like a fruit. The number 1 is not very interesting, about like 7 partly dried up, but when it is written "one" then it seems big and mysterious, ready to swallow everything up, the grandfather of numbers.

Perhaps you are surprised to read about numbers this way. They are usually treated much more dryly, because most mathematicians have forgotten about these pictures by the time they grow up. Writers might remember, but they are usually more interested in people, and think of numbers as impersonal and cold.

The larger numbers, to my mind, follow certain patterns. Usually odd numbers are darker than even numbers. Some odd numbers are wet, but all even numbers are dry. A two-digit number takes after its digits, but it also resembles its factors, the numbers that go into it evenly. For example, 92 seems like great heavy, smooth blocks. Now I think this is because 92 is a multiple of 4, which makes it squarish instead of round. But instead of being doughy, 92 is smooth and solid like 2, and heavy because of the 9. But 92 also makes me think of a movie called "The House on 92nd Street," and I used to live near there in Manhattan and seem to remember a house on 92nd Street made of heavy stone blocks.

All this depends on the cast of my mind. I rarely associate the numbers with colors. I incline more to textures—wet, dry, smooth, rough, heavy, light. When I was a child, I had no gift for art, but was fascinated by the way things felt to the touch. I used to run my hand over surfaces and learn the cool, restful feel of wood, the cold bite of metal, and the chill of glass.

Although I don't use colors, I do use light and dark. All numbers in the 20's seem flooded with sunlight. Numbers in the 30's are bumpy, in the 60's are very dry, and in the 70's are dark and mysterious. Numbers in the 40's don't seem at all doughy, like 4. It is the numbers ending in 4 that seem doughy, and the doughiest of all is 64, which is 4 × 4 × 4. The doughiness of 64 has to do with its having no odd factors. You can divide it by 2, and divide it again and again until you get all the way down to 1 and have nothing left to divide by, as if you rubbed a piece of dough between your hands until it had all blown away. Sixty-six is also dry and powdery, but it is gritty, for you know that if you divided it by 2 you would get stuck at dirty 33.

Some of the most beautiful numbers are multiples of 5. Twenty-five is thin and brilliant and round, but I prefer 75 with its dark, rich glow, like a deep ruby or a moonlit evening. The roundness of both numbers comes from 5 × 5, and the light or dark from 2 or 7. Specially intense effects are produced by a multiple of some number which also contains that number as a digit, as 36 contains 6 or 64 contains 4. For example, 16 is rich because of the 6 and strong like the roots of a tree because it is 4 × 4. But 96 = 16 × 6 is rich and strong and also has an extra coloring from the 9. Seventy-two = 12 × 6 has the bell-like quality of 12, but much deeper. One hundred and twenty-five is like 25, and 375 is like 75. The powers of 10 are like chiefs of different degrees. One hundred is a captain on horseback, and 1000 is a king, crusted with majesty.

Obviously, the multiples of 5 and 10 have special qualities only because we count by fives and tens. If you have studied the New Math, you know that we can just as well count by some other number, say by eights. Then the number 4 × 4 × 4, which I think of as all made of dough, would be written 100 and would be sitting on horseback as a chieftain!

In any case, the relation of a number to its factors is the same, no matter how we count. To my mind, multiples of 5 are round, of 2 are dry, of 4 are strong, of 7 are sad, of 10 are bland, of 3 are juicy. But some numbers, even large ones, have no factors—except themselves, of course, and 1. These are called prime numbers, because everything they are starts with themselves. They are original, gnarled, unpredictable, the freaks of the number world.


Pythagoras and Ramanujan

All this is just the opposite of arithmetic, which treats all numbers alike and reduces them to a system. People say that when you are in the army you are treated as if you were a number. No one cares who you are or what you are like, as long as you appear in the right place at the right time. But the reason we call that "being treated as a number" is that we have all learned, in arithmetic, to treat numbers as soldiers. Arithmetic is the gospel of those who are interested in hard, cold facts, who don't care whether 7 is a prime number, but who know that 7 dollars are better than 6, and 8 better than 7.

Consider the fascinating numbers 64, 65, 63. Sixty-four is the smallest number, after 1, that is both a square and a cube (64 = 82 = 43). Sixty-five is the smallest number which is the sum of two squares in two different ways (65 = 82 + 12 = 42 + 72). Sixty-three is the smallest number which is not prime, not a cube, not 1 more than a multiple of 4, and not a multiple of the sum of two squares! (For example, 15 is a multiple of 5 = 22 + 12; 21 is 1 more than 5 × 4; 27 is the cube of 3.)

In arithmetic these three numbers are lined up like soldiers on parade. Sixty-four is not allowed to flaunt its special traits but must keep step, in front of 65 and after 63. In fact, it is important, if you want to be good at arithmetic, to be able to handle all numbers with the same regular rhythm, just as a first baseman tries to cultivate the same rhythm for the hard throws as for the easy ones. So when we study the special properties of individual numbers, we are venturing far from arithmetic.

This venture, "useless" though it may seem, has attracted men for at least twenty-five centuries. In ancient Greek times, there was a school of numerology, headed by Pythagoras, which was more of a religious cult than an academic institution. The Pythagoreans were not only interested in numbers, they also believed that everything in the world is made out of numbers, so that if you understand numbers you can understand everything. Nowadays, of course, there are many fields of study that require a knowledge of mathematics, but the Pythagoreans meant something different. They thought that each number has its special qualities. For example, 4 signifies justice and 3 signifies power. They also believed that the qualities of a physical object are determined by the numbers that go into its construction. That is, a house built with 482 bricks would have the special qualities of 482. However, no one can be absolutely sure what the Pythagoreans believed, because they considered their ideas to be religious mysteries, which could not be revealed to outsiders. So they did not write any books about their system, and all we know of them comes from other Greeks who were not in the cult, but only wrote down what they had heard about it.


Pythagoras

At any rate, the followers of Pythagoras probably had a table of what they considered to be the qualities of numbers. When a number is the smallest to possess some property, it becomes interesting. We must then say that unless we count as far as that number, we cannot know all there is to know about numbers. For example, if we had never counted as far as 64, we might think that it was impossible for any number besides 1 to be both a square and a cube. When we reached 64, we would see that we had been mistaken. Thus, 64 is an interesting number; it shows us a new possibility of numbers.

Now, you might think that most numbers are uninteresting, but one can easily be fooled. There is a story about the number 1729 and the brilliant Indian mathematician Ramanujan, who died in 1920. Once when Ramanujan was sick, a friend came to visit him in a taxi with the number 1729 on it. The friend could not think of anything interesting about this number, although he tried during the whole trip. When he arrived, he told Ramanujan about the uninteresting number. Ramanujan was famous for his intimate knowledge of numbers. He said immediately, "Why, 1729 is the smallest number which is the sum of two cubes in two different ways!" (One way he had in mind was 1729 = 1728 + 1 = 123 + 13. Can you find the other way?)

But even Ramanujan might not be able to think of anything interesting about, let us say, a six-digit number. There is a very clever argument, though, for the proposition that every number is interesting in some way. Suppose that you make a list of interesting numbers, and that it includes all numbers up to 113789, but that 113789 is uninteresting. Then 113789 is the smallest uninteresting number, which is very interesting—so 113789 should be included on the list. But then 113789 is no longer uninteresting, so that it cannot be called the smallest uninteresting number. But then it should not be on the list. This is a paradox, and it shows that there cannot be any smallest uninteresting number (for such a number would be interesting), and therefore there cannot be any uninteresting number.

Now, this argument shouldn't be taken too seriously, because it assumes that every number is either interesting or not. Actually there are degrees, and as you count higher the numbers tend to get less interesting. Just which numbers are more interesting, and which less, is something about which there is no absolute rule. But I should like to see Ramanujan's own table of interesting properties of the first thousand numbers.

After the death of Pythagoras, his followers split into two groups. One group was mainly interested in worshipping numbers, and the other in studying them. The more religious division didn't last long, and it is easy to see why. It depended too much on controversial things. For example, I have said before that to me the odd numbers appear darker than the even. But in the Chinese philosophy of Yang and Yin, the odd numbers represent light, and the even numbers, darkness. There is no way to prove that either opinion is right, and if any two students of Pythagoras disagreed about such things, they could only appeal to the master for a decision. Once he was dead, there was no way to settle differences of opinion, and the cult gradually broke up.

It was the other followers of Pythagoras, those interested in exploring numbers as they are, who began something which did last and is still going on. They discovered some real facts about numbers, which don't depend on anyone's opinion. These facts are called theorems, from a Greek word meaning "look." Let's examine the difference between a theorem and a theory, and between the scientific and the popular meaning of the word "theory."


Only a Theory

In mathematics, a theorem is a statement that can be proved. For example, a simple theorem about numbers is that the product of two consecutive numbers is always even. (Examples: 3 × 4 = 12, 8 × 9 = 72, 22 × 23 = 506.) The proof is as follows. Of two consecutive numbers, one is always odd and one even. The product of an odd and an even number is always even, because the even factor is "2 × " some other number. Thus, if m is odd and n is even, then n = 2k, and mn = 2mk. So mn is even, being twice mk.

The proof of a theorem is like a legal argument. When you write a proof, you must imagine that there is a lawyer working against you who claims that the theorem is not true and who will raise objections whenever he can. If we had stopped after asserting that the product of an odd and an even number is always even, the opposing lawyer might have said, "Maybe not." So we backed up the assertion with an argument.

A theorem always consists of a statement and a proof. Sometimes a theorem is stated without proof, but the proof must exist or the theorem is not a theorem. The statement without the proof is like a menu without the dinner. When you read the menu, you think about the items and build up an appetite. If no one comes to serve you the dinner, you may be hungry enough to cook it yourself. But a menu is not a recipe, and there is nothing in the statement of a theorem to advise you how to construct the proof. Once in the kitchen, you are on your own. Here are the statements of two theorems; the proofs will be "served" later in the book.

1. The square of any odd number is 1 more than some multiple of 8. (Examples: 52 = 1 + 8 × 3, 92 = 1 + 8 × 10.)

2. The sum of the first so many cubes is always a square. (Examples: 13 + 23 + 33 = 1 + 8 + 27 = 62, 13 + 23 + 33 + 43 = 1 + 8 + 27 + 64 = 102.)

In everyday language, a theory is just the opposite of a. theorem. It is a statement that has not been proved and may even be false. Thus, Inspector Higginbottom has a theory that Spangley is the murderer, but he may be wrong. My wife has a theory that no man ever takes cream without sugar in his coffee, but she cannot prove it. In the popular sense, the word gives an impression of uncertainty and incompleteness; we say, "It is only a theory."

There is, however, a second meaning in the word theory. In this, the scientific meaning, a theory is not a statement at all, but a whole body of knowledge dealing with the ideas and rules behind some activity. If you have studied music and have taken a course in theory, you have learned about the relations of notes and chords and the rules of harmony. This is a very definite and precise subject, and there is nothing uncertain about it. Nor does it make sense to say that musical theory may or may not be true, because it is a whole subject and not a single statement. It is called "theory" because it is abstract and serves as background for the actual listening to or writing of music. In the same way, the "theory of perspective" is not a piece of guesswork but a definite body of rules and reasons which a draftsman (someone who draws) must master in order to draw pictures that give an impression of depth.

In science and mathematics, a statement that has not been proved is called not a theory, but an hypothesis. Hypo is the Greek word for "under," and thesis for "put," which makes sense because sub is Latin for "under" and pose for "put." So an hypothesis is something you suppose, something you put at the bottom of your mind in order to see what can be built on it.

Sometimes the scientific and popular meanings are combined. The name theory is often given to a whole body of reasoning which rests on some unproven assumptions. Examples are Darwin's theory of evolution, Karl Marx's theory of capitalist economies, and Einstein's theory of relativity. This kind of theory resembles an hypothesis in that it can be believed or disbelieved, and evidence may be brought to support it. If enough evidence is brought against it, the theory may be overthrown, like the phlogiston theory of chemistry around 1800 or the caloric theory of heat around 1850. But a theory, in this sense, is more than an hypothesis, for it is not a mere statement but a whole body of reasoning. The theory of evolution is not just the assertion that animals have developed over many generations from other animals; it is the whole study of how they have developed, how fast, when, where, and why. Even Inspector Higginbottom's theory about the murderer is usually more than a bald hypothesis; it is a chain of reasoning that explains why Spangley wanted to kill his uncle, who the mysterious woman was, and how Spangley got out through the locked door.

The mixing of meanings sometimes generates false impressions about theories. The theory of relativity, when proposed in 1905, consisted mainly of a few bold hypotheses and some interesting consequences. It was appropriate, then, to say "It is only a theory," because the hypotheses were unsure. But, since then, the basic hypotheses and many striking consequences have been confirmed by experiment. (This does not make them theorems, because a theorem can be proved without experiment, just by thinking.) The theory of relativity is no longer uncertain; it is quite as firmly established as any other branch of physics. A physicist studies the theory of relativity just as a radio engineer studies the theory of electric circuits. Calling it a theory does not mean that you doubt relativity, or doubt the existence of electric circuits. It just means that you have a lot to learn before you can build a radio. So it is no longer appropriate to say that the theory of relativity is "only a theory," as one might have said in 1910. Then it was a theory in the popular sense; now it is a theory in the scientific sense.


(Continues...)

Excerpted from AN ADVENTURER'S GUIDE TO NUMBER THEORY by RICHARD FRIEDBERG. Copyright © 1994 Richard Friedberg. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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Table of Contents

1. Seven jogged my elbow
2. On a clear day you can count forever
3. What goes in must come out
4. Arithmetica
5. A narrow margin
6. When the clock strikes thirteen
7. Hard nuts
8. A new wind
9. Roots go deep
10. Proofs of a pudding
  Further Reading
  Table of Theorems
  Index of Mathematicians
  Appendix I
  Appendix II
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  • Anonymous

    Posted January 1, 2007

    A conversational introduction with a historical perspective.

    Friedberg's text is a flawed introduction to number theory. The material is introduced through problems that motivate the results that Friedberg discusses. These results include Euclid's theorem that there are infinitely many prime numbers, the use of the sieve of Eratosthenes to find prime numbers less than the square root of a positive integer n, the Fundamental Theorem of Arithmetic, perfect and amicable numbers, Pythagorean triples, modular arithmetic, factoring numbers of the form x^2 + ny^2, and the Law of Quadratic Reciprocity. Friedberg, who writes in a conversational tone, ably links these topics together and places them in historical perspective. However, there are better introductions to the subject. This text has no formal exercises, so you do not have an opportunity to reinforce what you are learning. It is also a poor reference because definitions, theorems, and proofs are stated within paragraphs, the whimsical chapter titles do not convey what topics are covered, and there is no subject index to help you find the definitions and theorems that are buried within the paragraphs. What I found most disturbing was his assertion is that 1 is a prime number, which would eliminate unique factorization from the Fundamental Theorem of Arithmetic.

    1 out of 1 people found this review helpful.

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