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Prime and composite numbers

Prime and composite numbers

Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. This means they can only be divided evenly by 1 and the number itself. Examples include 2, 3, 5, 7, and 11.

Composite numbers, on the other hand, are natural numbers greater than 1 that have more than two positive divisors. This means they can be divided evenly by 1, themselves, and at least one other number. Examples include 4 (divisible by 1, 2, and 4), 6 (divisible by 1, 2, 3, and 6), and 8 (divisible by 1, 2, 4, and 8).

In summary, prime numbers are indivisible by any other numbers except for 1 and themselves, while composite numbers have additional divisors. The number 1 is neither prime nor composite.

Part 1: Prime numbers

Prime numbers are numbers that have only 2 factors: 1 and themselves. For example, the first 5 prime numbers are 2, 3, 5, 7, and 11. By contrast, numbers with more than 2 factors are call composite numbers.

Here are the key points to learn when studying prime numbers:

  1. Definition: Prime numbers are natural numbers greater than 1 that have no divisors other than 1 and themselves.

  2. Examples: The first few prime numbers include 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

  3. Unique Properties:

    • The number 2 is the only even prime number; all other even numbers are composite.
    • Prime numbers cannot be formed by multiplying two smaller natural numbers.
  4. Fundamental Theorem of Arithmetic: Every integer greater than 1 can be uniquely factored into prime numbers.

  5. Distribution: Prime numbers become less frequent as numbers get larger, but there are infinitely many primes (Euclid's Theorem).

  6. Tests for Primality: Various methods are used to determine if a number is prime, including trial division, the Sieve of Eratosthenes, and more advanced algorithms for large numbers.

  7. Applications: Prime numbers are crucial in number theory, cryptography, computer science, and various algorithms.

  8. Famous Conjectures: Includes the Goldbach Conjecture and the Twin Prime Conjecture.

  9. Patterns and Predictions: While there’s no simple formula for generating all prime numbers, researchers study their distribution and potential patterns.

  10. Prime Number Theorems: Various mathematical theorems explore the behavior and distribution of primes.

Understanding these key points will provide a solid foundation in the study of prime numbers.

Part 2: Recognizing prime and composite numbers

Can you recognize the prime numbers in this group of numbers? Which are prime, composite, or neither?

Here are the key points to learn when studying "Recognizing prime and composite numbers":

  1. Definitions:

    • Prime Numbers: A natural number greater than 1 that has no divisors other than 1 and itself. It has exactly two distinct positive divisors.
    • Composite Numbers: A natural number greater than 1 that has more than two distinct positive divisors.
  2. Identifying Numbers:

    • A prime number cannot be formed by multiplying two smaller natural numbers.
    • A composite number can be expressed as a product of two or more smaller natural numbers.
  3. Examples:

    • Prime: 2, 3, 5, 7, 11, 13, 17, etc.
    • Composite: 4, 6, 8, 9, 10, 12, etc.
  4. Special Case:

    • The number 1 is neither prime nor composite.
  5. Testing for Primality:

    • Efficiently test if a number is prime by checking for divisibility only up to its square root.
  6. Sieve of Eratosthenes:

    • A method for listing all primes up to a certain limit by systematically marking the multiples of each prime starting from 2.
  7. Applications:

    • Understanding prime and composite numbers is crucial in number theory, cryptography, and various mathematical theories.
  8. Properties:

    • The only even prime number is 2; all other even numbers are composite.
    • Large primes are rare, while composites are more common.

By focusing on these points, one can build a strong foundation in recognizing and understanding prime and composite numbers.