1.3 Fundamental theorem of arithmetic and its applications

The Fundamental Theorem of Arithmetic is a cornerstone of number theory. It states that every positive integer has a unique prime factorization, providing a foundation for many mathematical concepts and applications.

This theorem's power extends to various areas of number theory. It enables efficient calculations of greatest common divisors and least common multiples, and plays a crucial role in defining multiplicative functions and solving complex mathematical problems.

Prime Factorization and Unique Factorization

Understanding Prime Factorization

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• Prime factorization decomposes a positive integer into a product of prime factors
• Every positive integer greater than 1 can be expressed as a unique product of prime numbers
• Process involves dividing the number by the smallest prime factor repeatedly until reaching 1
• Prime factors are typically written in ascending order
• Notation uses exponents to represent repeated prime factors ($216 = 2^3 \times 3^3$)

The Fundamental Theorem of Arithmetic

• States that every positive integer has a unique prime factorization
• Provides the foundation for many number theory concepts and proofs
• Ensures consistency in representing numbers as products of primes
• Applies to all positive integers, including prime numbers themselves
• Prime numbers have a trivial prime factorization (the number itself)

Applications of Prime Factorization

• Simplifies complex calculations by breaking numbers into manageable parts
• Facilitates finding divisors and common factors of large numbers
• Enables efficient computation of greatest common divisors and least common multiples
• Plays a crucial role in various cryptographic algorithms (RSA encryption)
• Helps in solving Diophantine equations and other number theory problems

Multiplicative Functions in Number Theory

• Multiplicative functions preserve multiplication: $f(ab) = f(a)f(b)$ for coprime a and b
• Examples include Euler's totient function and the Möbius function
• Utilize the unique prime factorization of integers in their definitions
• Often defined recursively based on prime factorizations
• Frequently appear in sum and product formulas in analytic number theory

Greatest Common Divisor and Least Common Multiple

Defining and Computing GCD

• Greatest Common Divisor (GCD) represents the largest positive integer that divides both numbers without a remainder
• Denoted as $\gcd(a,b)$ for integers a and b
• Can be computed using prime factorizations of the numbers
• Alternative method uses the Euclidean algorithm for efficient calculation
• GCD of more than two numbers can be found by applying the operation pairwise

Properties and Applications of LCM

• Least Common Multiple (LCM) is the smallest positive integer divisible by both numbers
• Denoted as $\text{lcm}(a,b)$ for integers a and b
• Calculated using the prime factorizations of the numbers
• Relates to GCD through the formula: $\text{lcm}(a,b) \times \gcd(a,b) = |ab|$
• Useful in various applications (finding common denominators in fraction arithmetic)

The Euclidean Algorithm and Its Extensions

• Efficient method for computing GCD without factoring the numbers
• Based on the principle that $\gcd(a,b) = \gcd(b, a \bmod b)$
• Involves repeated division with remainder until the remainder becomes zero
• Can be extended to find coefficients in Bézout's identity
• Forms the basis for solving linear Diophantine equations
• Generalizes to polynomials and other algebraic structures

Coprime Numbers

Properties of Coprime Integers

• Coprime numbers (relatively prime) have a greatest common divisor of 1
• Two numbers are coprime if their prime factorizations have no common prime factors
• Any two consecutive integers are always coprime
• The probability of two randomly chosen integers being coprime approaches $\frac{6}{\pi^2}$ (about 0.6079)
• Coprimality is preserved under addition and subtraction (if a and b are coprime, a+b and a-b are coprime to b)

Applications of Coprimality in Number Theory

• Fundamental in the definition and properties of multiplicative functions
• Essential in modular arithmetic and the Chinese Remainder Theorem
• Used in the formulation of Euler's totient function
• Plays a key role in various cryptographic protocols (selection of public and private keys in RSA)
• Simplifies fraction arithmetic (when numerator and denominator are coprime, the fraction is in its lowest terms)

Coprimality and Mathematical Proofs

• Often used as a hypothesis in number theory theorems and proofs
• Facilitates the use of the Euclidean algorithm in various contexts
• Allows for the application of Bézout's identity in solving linear Diophantine equations
• Helps in proving properties of congruences and modular arithmetic
• Used in the proof of the Fundamental Theorem of Arithmetic itself