# The Ultimate Guide to Elementary Number Theory by Burton 7th Edition PDF.zip: Concepts, Examples and Exercises

## Elementary Number Theory by Burton 7th Edition PDF.zip: A Must-Have Resource for Math Enthusiasts

If you are looking for a comprehensive and accessible introduction to the fascinating field of number theory, you might want to check out Elementary Number Theory by Burton 7th Edition PDF.zip. This book is written for the one-semester undergraduate number theory course taken by math majors, secondary education majors, and computer science students. It covers the main topics of classical number theory, such as divisibility, primes, congruences, Fermat's theorem, Euler's theorem, quadratic reciprocity, perfect numbers, Fibonacci numbers, and more. It also provides some historical background that shows the subject's evolution from antiquity to recent research.

## elementary number theory burton 7th edition pdf.zip

Elementary Number Theory by Burton 7th Edition PDF.zip is a well-written and engaging text that explains the concepts and proofs in a clear and concise manner. It also includes many examples, exercises, and applications that illustrate the relevance and beauty of number theory. The book is suitable for both self-study and classroom use, as it contains hints and answers to selected exercises at the end of each chapter. The book also has a companion website that offers additional resources, such as online quizzes, supplementary material, and links to related websites.

Elementary Number Theory by Burton 7th Edition PDF.zip is a valuable resource for anyone who wants to learn more about the theory and practice of number theory. It is also a great preparation for more advanced courses and research in number theory and related fields. Whether you are a student, a teacher, or a hobbyist, you will find this book to be informative, enjoyable, and inspiring.

In this article, we will give you an overview of some of the main topics covered in Elementary Number Theory by Burton 7th Edition PDF.zip. We will also highlight some of the features and benefits of this book, as well as how you can download it for free.

## Divisibility Theory in the Integers

One of the fundamental concepts in number theory is divisibility. This means that one integer can be divided by another integer without leaving a remainder. For example, 12 is divisible by 3, but not by 5. Divisibility theory deals with the properties and consequences of divisibility, such as the greatest common divisor, the least common multiple, the Euclidean algorithm, and the Fundamental Theorem of Arithmetic. The Fundamental Theorem of Arithmetic states that every positive integer can be written as a product of prime numbers in a unique way, up to the order of the factors. For example, 60 = 2 x 2 x 3 x 5 = 2 x 3 x 2 x 5 = 5 x 2 x 2 x 3, and so on.

## Primes and Their Distribution

A prime number is a positive integer that has exactly two positive divisors: itself and 1. For example, 2, 3, 5, 7, 11, and 13 are prime numbers. Primes are the building blocks of all integers, as we saw in the Fundamental Theorem of Arithmetic. Primes also have many interesting and mysterious properties, such as their patterns, their density, their distribution, and their conjectures. For example, one of the oldest and most famous conjectures in number theory is the Goldbach Conjecture, which states that every even integer greater than 2 can be written as the sum of two primes. For example, 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, and so on.

## The Theory of Congruences

A congruence is a relation between two integers that says that they have the same remainder when divided by a fixed positive integer called the modulus. For example, 17 and 5 are congruent modulo 12, because they both have a remainder of 5 when divided by 12. We write this as 17 5 (mod 12). Congruences are useful for simplifying calculations and solving equations involving integers. They also have applications in cryptography, coding theory, and modular arithmetic. Modular arithmetic is a system of arithmetic where numbers wrap around after reaching a certain value called the modulus. For example, in modulo 12 arithmetic, we have that 10 + 5 = 3 (mod 12), because adding 5 to 10 gives us a number that is equivalent to

3 modulo

12.

## Fermat's Theorem and Euler's Theorem

Another important topic in number theory is the study of modular exponentiation. This means raising a number to a power and then taking the remainder when divided by a modulus. For example, 2^3 = 8, and 8 mod 5 = 3. Modular exponentiation has many applications in cryptography, such as the RSA algorithm and the Diffie-Hellman key exchange. Two of the most famous results in modular exponentiation are Fermat's theorem and Euler's theorem. Fermat's theorem states that if p is a prime number and a is any integer that is not divisible by p, then a^(p-1) 1 (mod p). For example, if p = 7 and a = 3, then 3^6 1 (mod 7). Euler's theorem is a generalization of Fermat's theorem that works for any modulus that is not necessarily prime. Euler's theorem states that if n is any positive integer and a is any integer that is relatively prime to n, then a^(φ(n)) 1 (mod n), where φ(n) is the Euler totient function that counts the number of positive integers less than n that are relatively prime to n. For example, if n = 10 and a = 3, then φ(10) = 4 and 3^4 1 (mod 10).

## Quadratic Reciprocity and Sums of Squares

A quadratic residue is an integer that can be written as the square of another integer modulo a given modulus. For example, 4 is a quadratic residue modulo 7, because 4 2^2 (mod 7). A quadratic non-residue is an integer that cannot be written as the square of another integer modulo a given modulus. For example, 3 is a quadratic non-residue modulo 7, because there is no integer x such that x^2 3 (mod 7). The quadratic reciprocity law is a remarkable theorem that relates the quadratic residues and non-residues of two different moduli. It states that if p and q are distinct odd prime numbers, then p is a quadratic residue modulo q if and only if q is a quadratic residue modulo p, except when both p and q are congruent to 3 modulo 4, in which case the opposite holds. For example, if p = 5 and q = 7, then both p and q are congruent to 1 modulo 4, so p is a quadratic residue modulo q if and only if q is a quadratic residue modulo p. Indeed, we have that 5 (3)^2 (mod

7) and

7 (2)^2 (mod

5).

A related problem in number theory is to determine which integers can be written as the sum of two or more squares. For example, we can write 13 = 2^2 + 3^2 or 25 = 0^2 + 5^2 or

25 = (-5)^2 + (-0)^2

as sums of two squares. However, we cannot write

7 or

11 as sums of two squares. A famous result by Fermat states that an odd prime number can be written as the sum of two squares if and only if it is congruent to

1 modulo

4. For example,

5

1 (mod

4) and

5 = (-1)^2 + (-2)^2 or

5 = (-2)^2 + (-1)^2 or

5 = (1)^2 + (2)^2 or

5 = (2)^2 + (1)^2 as sums of two squares. However,

7

3 (mod

4) and

7 cannot be written as the sum of two squares. b99f773239

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