Glossary

5.1 Divisibility and Prime Numbers

4 min read•august 12, 2024

Number theory forms the backbone of cryptography. Divisibility and prime numbers are fundamental concepts that unlock the secrets of integers. These ideas lay the groundwork for understanding how numbers interact and form the basis for secure communication systems.

Prime factorization, greatest common divisors, and least common multiples build on these basics. These tools allow us to manipulate numbers in powerful ways, enabling the creation of encryption algorithms that keep our digital world safe and secure.

Divisibility and Number Types

Fundamental Concepts of Divisibility and Prime Numbers

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  • Divisibility occurs when one integer divides another without a remainder
    • Expressed as "a divides b" or "b is divisible by a"
    • Mathematically written as a∣ba | b if there exists an integer kk such that b=akb = ak
  • Prime numbers possess only two factors: 1 and themselves
    • Examples include 2, 3, 5, 7, 11, 13, 17, 19
    • 2 stands as the only even
  • Composite numbers have more than two factors
    • Can be expressed as the product of two smaller positive integers
    • Numbers like 4, 6, 8, 9, 10, 12 fall into this category
  • states every positive integer greater than 1 can be uniquely represented as a product of prime numbers
    • Also known as the unique factorization theorem
    • Allows for the expression of any number as a product of prime factors

Prime Factorization and Its Applications

  • Prime factorization involves breaking down a number into its prime factors
    • Process of finding the set of prime numbers that multiply together to form the original number
    • Useful in various mathematical operations and problem-solving
  • Steps to perform prime factorization:
    1. Start dividing the number by the smallest prime number that divides it evenly
    2. Continue dividing the quotient by prime numbers until the quotient becomes 1
    • 60 can be factored as 2∗2∗3∗52 * 2 * 3 * 5 or 22∗3∗52^2 * 3 * 5
  • Applications of prime factorization:
    • Simplifying fractions
    • Finding least common multiples and greatest common divisors
    • Solving certain types of Diophantine equations

Greatest Common Divisor and Least Common Multiple

Understanding and Calculating GCD

  • (GCD) represents the largest positive integer that divides two or more integers without a remainder
    • Also referred to as the Greatest Common Factor (GCF) or Highest Common Factor (HCF)
    • For integers a and b, denoted as GCD(a,b)GCD(a,b) or gcd⁡(a,b)\gcd(a,b)
  • Methods to find GCD:
    1. Listing factors: List all factors of each number and identify the greatest common one
    2. Prime factorization: Factor each number and multiply the common prime factors
    3. Euclidean algorithm: An efficient method for larger numbers
  • Euclidean algorithm steps:
    1. Divide the larger number by the smaller one
    2. Replace the larger number with the smaller number and the smaller number with the remainder
    3. Repeat until the remainder is zero
    • The last non-zero remainder is the GCD
  • Properties of GCD:
    • gcd⁡(a,b)=gcd⁡(b,a)\gcd(a,b) = \gcd(b,a)
    • gcd⁡(a,0)=∣a∣\gcd(a,0) = |a| for any non-zero a
    • gcd⁡(a,1)=1\gcd(a,1) = 1 for any integer a

Least Common Multiple and Its Calculation

  • (LCM) is the smallest positive integer that is divisible by two or more given integers
    • Denoted as LCM(a,b)LCM(a,b) or lcm(a,b)lcm(a,b) for integers a and b
  • Methods to find LCM:
    1. Listing multiples: List multiples of each number and identify the smallest common one
    2. Prime factorization: Factor each number and multiply all prime factors using the highest power in which they occur
    3. Using GCD: LCM(a,b)=∣a∗b∣GCD(a,b)LCM(a,b) = \frac{|a * b|}{GCD(a,b)}
  • Relationship between LCM and GCD:
    • The product of LCM and GCD of two numbers equals the product of the numbers themselves
    • Expressed as LCM(a,b)∗GCD(a,b)=∣a∗b∣LCM(a,b) * GCD(a,b) = |a * b|
  • Applications of LCM:
    • Finding common denominators in fraction addition and subtraction
    • Solving scheduling and synchronization problems
    • Determining when periodic events will coincide

Prime Number Generation

Sieve of Eratosthenes and Prime Number Properties

  • serves as an ancient and efficient algorithm for finding all prime numbers up to a given limit
    • Named after the ancient Greek mathematician Eratosthenes of Cyrene
  • Steps of the Sieve of Eratosthenes:
    1. Create a list of consecutive integers from 2 to the desired limit
    2. Start with the first number in the list (2) and mark all its multiples as composite
    3. Move to the next unmarked number and repeat step 2
    4. Continue until reaching the square root of the limit
  • Efficiency of the sieve:
    • Time complexity: O(nlog⁡log⁡n)O(n \log \log n)
    • Space complexity: O(n)O(n)
  • Properties of prime numbers:
    • : There are infinitely many prime numbers
    • Distribution of primes becomes less dense as numbers increase
    • Prime Number Theorem describes the asymptotic distribution of prime numbers
  • Applications of prime numbers:
    • Cryptography and secure communication (RSA algorithm)
    • Hash functions and data structures
    • Random number generation

Key Terms to Review (18)

Aks primality test: The AKS primality test is a deterministic algorithm used to determine whether a number is prime. It was introduced in 2002 by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, and it represents a significant breakthrough because it runs in polynomial time. This test connects to the broader understanding of prime numbers and their properties, particularly in relation to divisibility and factorization.
Composite Number: A composite number is a positive integer that has at least one positive divisor other than one or itself, meaning it can be divided evenly by numbers other than just 1 and itself. This characteristic distinguishes composite numbers from prime numbers, which only have two distinct positive divisors: 1 and the number itself. Composite numbers play a crucial role in number theory, particularly in understanding factors and divisibility.
Divisibility Rules: Divisibility rules are a set of shortcuts that help determine whether one number is divisible by another without performing full division. These rules simplify calculations and are especially useful when working with larger numbers or in problems involving factors and multiples. Understanding these rules can also help in identifying prime numbers and their properties, as divisibility is fundamental to number theory.
Divisor: A divisor is a number that divides another number without leaving a remainder. This concept is essential in understanding how integers interact with one another and plays a significant role in identifying factors, simplifying fractions, and working with prime numbers. Recognizing the relationship between divisors and multiples helps to establish foundational principles in number theory.
Euclid: Euclid was an ancient Greek mathematician, often referred to as the 'Father of Geometry.' He is best known for his work 'Elements,' which is a comprehensive compilation of the knowledge of geometry at his time. His axiomatic approach laid the groundwork for modern mathematics, particularly in the study of divisibility and prime numbers, by formalizing definitions and proving theorems in a systematic way.
Euclid's Theorem: Euclid's Theorem states that there are infinitely many prime numbers. This fundamental result in number theory was first presented by the ancient Greek mathematician Euclid in his work 'Elements.' The theorem establishes the significance of prime numbers in mathematics and lays the groundwork for understanding their distribution.
Fermat's Little Theorem: Fermat's Little Theorem states that if 'p' is a prime number and 'a' is any integer not divisible by 'p', then $$a^{p-1} \equiv 1 \mod p$$. This theorem provides a foundation for understanding properties of prime numbers and their relationship to modular arithmetic, offering insights into the behavior of integers under division by primes and the simplifications possible in calculations involving large powers.
Fundamental Theorem of Arithmetic: The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be uniquely represented as a product of prime numbers, up to the order of the factors. This means that no matter how you factor a number into primes, you will always arrive at the same set of prime factors, which highlights the central role of prime numbers in the structure of integers. This theorem serves as a foundational concept in number theory, establishing a clear connection between divisibility and primes.
Greatest common divisor: The greatest common divisor (GCD) of two or more integers is the largest positive integer that divides each of the numbers without leaving a remainder. Understanding the GCD is crucial for various applications in number theory, such as simplifying fractions and finding least common multiples. The concept ties closely to divisibility and prime numbers, as prime factorization can help determine the GCD by identifying common factors among the given integers.
Infinitude of Primes: The infinitude of primes is the theorem that states there are infinitely many prime numbers, meaning that no matter how far you go along the number line, you can always find more primes. This concept highlights the endless nature of prime numbers and serves as a cornerstone in number theory, illustrating the richness and complexity of the integers. The proof of this theorem is often attributed to Euclid, who demonstrated it over two millennia ago, and it continues to be a foundational aspect of understanding prime distribution.
Least common multiple: The least common multiple (LCM) of two or more integers is the smallest positive integer that is divisible by each of those integers. Understanding LCM is crucial in operations involving fractions, as it helps to find a common denominator. Additionally, the LCM relates to prime factorization, which is fundamental in identifying the multiples of numbers and their relationships.
Mersenne primes: Mersenne primes are a special class of prime numbers that can be expressed in the form $$M_n = 2^n - 1$$, where $$n$$ is a positive integer. These primes are significant in number theory and have a deep connection to binary systems and perfect numbers, leading to important discoveries in mathematics. Identifying Mersenne primes is also crucial for understanding the distribution of prime numbers and their unique properties.
Multiple: A multiple is a number that can be expressed as the product of an integer and a given number, meaning it is evenly divisible by that number without leaving a remainder. This concept is closely tied to understanding divisibility, where identifying multiples can help determine factors, and it plays a critical role in working with prime numbers, as every integer has an infinite set of multiples. Recognizing multiples aids in various mathematical operations, including simplification and fraction reduction.
Pierre de Fermat: Pierre de Fermat was a French mathematician known for his contributions to number theory, particularly in the area of prime numbers and divisibility. His work laid the foundation for many concepts in modern mathematics, including Fermat's Little Theorem and Fermat's Last Theorem, which have significant implications in understanding the properties of prime numbers and their relationships.
Prime Number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. This unique property makes prime numbers the building blocks of whole numbers, as every integer greater than 1 can be expressed as a product of prime numbers. They play a crucial role in number theory, particularly in the study of divisibility and the fundamental theorem of arithmetic.
Sieve of eratosthenes: The sieve of Eratosthenes is an ancient algorithm used to find all prime numbers up to a specified integer. It works by iteratively marking the multiples of each prime number starting from 2, allowing for the efficient identification of prime numbers within a given range. This method highlights the concept of divisibility and serves as a foundational technique in number theory for understanding how primes are distributed.
Transitive Property of Divisibility: The transitive property of divisibility states that if one integer is divisible by a second integer, and the second integer is divisible by a third integer, then the first integer is also divisible by the third integer. This property highlights a fundamental relationship among integers and their divisibility, emphasizing how divisors can be linked through their multiples.
Trial division: Trial division is a simple algorithm used to determine whether a number is prime by testing divisibility against all integers up to the square root of that number. This method leverages the principle that if a number has a divisor larger than its square root, the corresponding factor must be smaller, making it efficient for identifying prime numbers. It plays a vital role in the study of divisibility and primes, as it directly aids in establishing the primality of numbers.
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