Prime Numbers and Primality Testing

Definition of Prime Numbers

• A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers.
• A natural number greater than 1 that is not prime is called a composite number.

Characteristics of Prime Numbers

• A prime number can only be divided by 1 and itself.
• For example, 5 is prime because the only ways of writing it as a product are 1 × 5 or 5 × 1.
• The property of being prime is called primality.

Primality Testing

• A simple but slow method of checking primality is trial division, which tests whether a number is a multiple of any integer between 2 and its square root.
• Faster algorithms include the Miller–Rabin primality test and the AKS primality test.

Properties of Prime Numbers

• There are infinitely many primes, as demonstrated by Euclid around 300 BC.
• No known simple formula separates prime numbers from composite numbers.
• The distribution of primes within the natural numbers can be statistically modelled.

Prime Number Theorem

• The prime number theorem states that the probability of a randomly chosen large number being prime is inversely proportional to its number of digits, or its logarithm.

Applications of Prime Numbers

• Prime numbers are used in public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors.
• In abstract algebra, objects that behave like prime numbers include prime elements and prime ideals.

Historical Questions and Unsolved Problems

• Goldbach's conjecture states that every even integer greater than 2 can be expressed as the sum of two primes.
• The twin prime conjecture states that there are infinitely many pairs of primes that differ by two.
• These questions have spurred the development of various branches of number theory.

Definition of Prime Numbers

• A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers.
• A natural number greater than 1 that is not prime is called a composite number.

Characteristics of Prime Numbers

• A prime number can only be divided by 1 and itself.
• For example, 5 is prime because the only ways of writing it as a product are 1 × 5 or 5 × 1.
• The property of being prime is called primality.

Primality Testing

• A simple but slow method of checking primality is trial division, which tests whether a number is a multiple of any integer between 2 and its square root.
• Faster algorithms include the Miller–Rabin primality test and the AKS primality test.

Properties of Prime Numbers

• There are infinitely many primes, as demonstrated by Euclid around 300 BC.
• No known simple formula separates prime numbers from composite numbers.
• The distribution of primes within the natural numbers can be statistically modelled.

Prime Number Theorem

• The prime number theorem states that the probability of a randomly chosen large number being prime is inversely proportional to its number of digits, or its logarithm.

Applications of Prime Numbers

• Prime numbers are used in public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors.
• In abstract algebra, objects that behave like prime numbers include prime elements and prime ideals.

Historical Questions and Unsolved Problems

• Goldbach's conjecture states that every even integer greater than 2 can be expressed as the sum of two primes.
• The twin prime conjecture states that there are infinitely many pairs of primes that differ by two.
• These questions have spurred the development of various branches of number theory.

Definition of Prime Numbers

• A prime number is a positive integer that is divisible only by itself and 1.
• Prime numbers are greater than 1 and have exactly two distinct positive divisors: 1 and themselves.

Characteristics of Prime Numbers

• Prime numbers are always odd, except for the number 2, which is the only even prime number.

Examples of Prime Numbers

• 2, 3, 5, 7, 11, 13, ...

Non-Examples of Prime Numbers

• 1 is not prime because it has only one divisor: 1.
• 4 is not prime because it has more than two divisors: 1, 2, and 4.
• 6 is not prime because it has more than two divisors: 1, 2, 3, and 6.

Importance of Prime Numbers

• Prime numbers play a fundamental role in number theory and are used in many mathematical concepts, such as cryptography, algebra, and geometry.

Definition Of Prime Numbers

• A prime number is a positive integer divisible only by itself and 1.
• It is a natural number greater than 1 with no positive divisors other than 1 and itself.
• The smallest prime number is 2.

Identifying Prime Numbers

• Trial Division is a method to check if a number is prime by dividing it by all prime numbers less than or equal to its square root.
• Fermat's Little Theorem states that if p is prime, then a^p ≡ a (mod p) for any integer a.
• Miller-Rabin Primality Test is a probabilistic algorithm to determine if a number is prime or composite.

Prime Number Theorems

• Euclid's Theorem states that there are infinitely many prime numbers.
• The Prime Number Theorem approximates the distribution of prime numbers using the logarithmic integral function.

Prime Factorization

• Prime Factorization is the process of expressing a composite number as a product of prime numbers.
• The Fundamental Theorem of Arithmetic states that every positive integer can be expressed as a unique product of prime numbers, except for the order in which they are listed.
• Factor Trees are a graphical method to find prime factors.
• The Sieve of Eratosthenes is an algorithm to find all prime numbers up to a given number.
• Pollard's Rho Algorithm is a method to find prime factors of large numbers.