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A Friendly Introduction to Number Theory is an introductory undergraduate Pearson is thus providing this chapter free of charge for download as a PDF file. Full file at link-marketing.info Edition-Silverman-Solutions-Manual Table of Contents Chapter 1 What is Number. Chapter 1 What Is Number Theory? Exercises The first two numbers that are both squares and triangles are 1 and Find the next one and, if possible, the.

Download Sample Chapter. Try to prove that your observation is true for all primitive Pythagorean triples. Topology of Numbers by Allen Hatcher - Cornell University An introductory textbook on elementary number theory from a geometric point of view, as opposed to the strictly algebraic approach. You have successfully signed out and will be required to sign back in should you need to download more resources. But no one has yet proved any of these conjectures. Errata for the 3rd edition is also available.

Chapter 2: Pythagorean Triples Chapter 3: Pythagorean Triples and the Unit Circle Chapter 4: Divisibility and the Greatest Common Divisor Chapter 6: Is -1 a Square Modulo p?

It has come to my attention that some low-cost editions of my book have been printed with this chapter omitted and the subsequent chapters renumbered. Pearson is thus providing this chapter free of charge for download as a PDF file. Continued Fractions and Pell's Equation Chapter Exercises 1. The first two numbers that are both squares and triangles are 1 and Find the next one and, if possible, the one after that.

Can you figure out an efficient way to find triangular—square numbers? Do you think that there are infinitely many? Solution to Exercise 1.

The first three triangular—square numbers are 36, , and The first few pairs are 8, 6 , 49, 35 , , , , , and , The pattern for generating these pairs is quite subtle. Starting with 1, 1 and applying this rule repeatedly will actually give all triangular—square numbers.

Try adding up the first few odd numbers and see if the numbers you get satisfy some sort of pattern. Once you find the pattern, express it as a formula.

Give a geometric verification that your formula is correct. The sum of the first n odd numbers is always a square. The following pictures illustrate the first few cases, and they make it clear how the general case works.

The consecutive odd numbers 3, 5, and 7 are all primes. The only prime triplet is 3, 5, 7.

The reason is that for any three odd numbers, at least one of them must be divisible by 3. So in order for them all to be prime, one of them must equal 3. First we accumulate some data, which we list in a table.

But no one has yet proved any of these conjectures. The following two lines indicate another way to derive the formula for the sum of the first n integers by rearranging the terms in the sum. Fill in the details. You may need to consider the cases of odd n and even n separately.

Suppose first that n is even. For each of the following statements, fill in the blank with an easy-to-check crite- rion: Use the same sort of argument to show that either a or b must be a multiple of 3. Try to show that your guess is correct. Solution to Exercise 2. A flowchart of chapter dependencies is included in this edition.

Five basic steps are emphasized throughout the text to help readers develop a robust thought process: Experimentation Pattern recognition Hypothesis formation Hypothesis testing Formal proof RSA cryptosystem, elliptic curves, and Fermat's Last Theorem are featured , showing the real-life applications of mathematics.

New to This Edition. There are a number of major changes in the Fourth Edition. Many new exercises appear throughout the text. A flowchart giving chapter dependencies is included to help instructors choose the most appropriate mix of topics for their students. Content Updates There is a new chapter on mathematical induction Chapter Some material on proof by contradiction has been moved forward to Chapter 8.

It is used in the proof that a polynomial of degree d has at most d roots modulo p. In earlier editions, primitive roots were used for this proof.

The chapters on primitive roots Chapters 28—29 have been moved to follow the chapters on quadratic reciprocity and sums of squares Chapters 20— The new order allows the instructor to cover quadratic reciprocity first, and to omit primitive roots entirely if desired. Chapter 22 now includes a proof of part of quadratic reciprocity for Jacobi symbols , with the remaining parts included as exercises.

Quadratic reciprocity is now proved in full. Chapter 23 is significantly more difficult than the chapters that precede it, and it may be omitted without affecting the subsequent chapters. As an application of primitive roots , Chapter 28 discusses the construction of Costas arrays.

Number theory is a vast and sprawling subject, and over the years this book has acquired many new chapters. In order to keep the length of this edition to a reasonable size, Chapters 47, 48, 49, and 50 have been removed from the printed version of the book. These omitted chapters are freely available online at http: The online chapters are included in the index. What Is Number Theory? Pythagorean Triples 3. Pythagorean Triples and the Unit Circle 4. Divisibility and the Greatest Common Divisor 6.

Linear Equations and the Greatest Common Divisor 7.

Factorization and the Fundamental Theorem of Arithmetic 8. Congruences 9. Prime Numbers Counting Primes Mersenne Primes Mersenne Primes and Perfect Numbers