🔢 Prime Numbers Table
Generate a comprehensive table of prime numbers up to a specified limit or the first n primes using the Sieve of Eratosthenes algorithm, perfect for mathematics education and reference.
🔢 Generate Prime Numbers
Generate prime numbers using the Sieve of Eratosthenes algorithm:
🔢 Understanding Prime Numbers
Prime numbers are integers greater than 1 that have no positive divisors other than 1 and themselves. They are the building blocks of number theory and have fascinated mathematicians for centuries.
🧮 What is a Prime Number?
📐 Definition
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23...
⚡ Composite Numbers
Numbers greater than 1 that are not prime. They can be factored into smaller integers. Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16...
🎯 Special Case: 2
2 is the only even prime number. All other even numbers are composite because they are divisible by 2.
🔢 Special Case: 1
1 is neither prime nor composite. It has only one positive divisor (itself) and doesn't fit the definition of prime numbers.
⚙️ Sieve of Eratosthenes
📜 Ancient Algorithm
Invented by Greek mathematician Eratosthenes around 200 BC. Still one of the most efficient ways to find all primes up to a limit.
🔄 How It Works
1. Create list of numbers from 2 to n
2. Start with smallest number (2)
3. Mark all multiples as composite
4. Move to next unmarked number
5. Repeat until √n
⚡ Efficiency
Time complexity: O(n log log n)
Space complexity: O(n)
Very efficient for finding all primes up to n
Optimal for ranges up to ~10^7
💻 Implementation
Uses boolean array for marking
Crosses out multiples of each prime
Collects remaining unmarked numbers
Perfect for this calculator
🧮 Prime Number Properties
♾️ Infinite Count
Euclid proved there are infinitely many primes around 300 BC. This was one of the first proofs by contradiction in mathematics.
📊 Distribution
Prime numbers become less frequent as numbers get larger. The Prime Number Theorem describes this distribution pattern.
👯 Twin Primes
Pairs of primes differing by 2 (e.g., 11-13, 17-19, 29-31). It's conjectured there are infinitely many twin primes.
🔐 Cryptography
RSA encryption relies on the difficulty of factoring large composite numbers into their prime factors.
💡 Prime Number Fact: The largest known prime number has over 24 million digits and was discovered by the Great Internet Mersenne Prime Search (GIMPS) project. It's a Mersenne prime of the form 2^82,589,933 - 1.
🔬 Famous Prime Numbers
🎯 Applications of Prime Numbers
🔐 Cybersecurity
RSA encryption algorithm
Public-key cryptography
Digital signatures
Secure communications
💻 Computer Science
Hash table sizing
Pseudorandom number generation
Computer graphics
Algorithm optimization
📊 Statistics
Random sampling methods
Monte Carlo simulations
Quality control
Experimental design
🎵 Music Theory
Harmonic series
Just intonation
Frequency ratios
Scale construction
🧪 Testing Primality
🔍 Trial Division
Divide by all numbers up to √n
Simple but slow for large numbers
O(√n) time complexity
Works well for small n
🎯 Fermat's Test
Probabilistic primality test
Fermat's Little Theorem
Fast but not always accurate
Used in some applications
🧬 Miller-Rabin Test
Deterministic for small numbers
Very fast and reliable
Used in cryptographic applications
Modern standard method
🔢 AKS Primality Test
Deterministic and polynomial-time
Mathematically proven correct
Too slow for practical use
Theoretical breakthrough
🌟 Unsolved Problems
👯 Twin Prime Conjecture
Are there infinitely many twin primes?
One of the most famous unsolved problems
Related to the distribution of primes
Millennium Prize Problem
🎲 Goldbach Conjecture
Can every even number > 2 be written as sum of two primes?
Conjectured by Christian Goldbach in 1742
Verified up to very large numbers
Still unproven
📊 Riemann Hypothesis
Concerns the zeros of the Riemann zeta function
One of the seven Millennium Prize Problems
Implies bounds on prime gaps
Extremely important in number theory
🔄 Prime Gap Conjecture
How large can gaps between consecutive primes be?
Related to the distribution of primes
Recent breakthroughs by Zhang and Maynard
Still active area of research