Prime Number Calculator
Check primality, generate prime numbers, and analyze prime factors
Tip: Use different calculation modes for various prime number applications - from basic checks to advanced factorization.
Check if a Number is Prime
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Prime Number Properties
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Digit Length:
Distance to Next Prime:
Special Properties:
The Fascinating World of Prime Numbers
Fundamental Theorem
- Every integer greater than 1 is either prime or a product of primes
- Prime factorization is unique (up to ordering)
- Forms the foundation of number theory
Modern Applications
- Cryptography (RSA encryption)
- Computer security algorithms
- Hash functions and checksums
"The distribution of primes among integers is one of mathematics' deepest mysteries. Despite their simple definition, primes exhibit remarkably complex behavior that continues to challenge mathematicians."
Types of Prime Numbers
Twin Primes
Pairs of primes that differ by 2 (e.g., 3 and 5, 11 and 13). The Twin Prime Conjecture suggests there are infinitely many such pairs, though this remains unproven. The largest known twin primes have over 388,000 digits each.
Mersenne Primes
Primes of the form 2p-1 where p is itself prime. These are connected to perfect numbers and are often the largest known primes. The largest known prime as of 2023 is 282,589,933-1 with 24,862,048 digits.
Sophie Germain Primes
Primes p where 2p+1 is also prime. These are important in cryptography and number theory. They're named after French mathematician Sophie Germain who used them in her work on Fermat's Last Theorem.
Did You Know? The Prime Number Theorem describes the asymptotic distribution of primes, showing they become less frequent as numbers grow larger, with primes near n having average spacing of about ln(n).
Frequently Asked Questions
How does this calculator check for primality?
For numbers up to 1012, we use deterministic tests. For larger numbers (up to 15 digits), we use probabilistic tests (Miller-Rabin) which are extremely accurate for practical purposes. True primality proofs for very large numbers require specialized algorithms.
What's the largest number this calculator can handle?
Primality checking works up to 15-digit numbers (999 trillion). Prime generation handles ranges up to 100 million. Factorization works best for numbers up to 12 digits, though some larger numbers may work depending on their factors.
Why is prime factorization important?
Factorization is fundamental to number theory and forms the basis of modern cryptography. The difficulty of factoring large numbers is what makes RSA encryption secure. It's also essential for simplifying fractions and solving Diophantine equations.
Are there infinitely many prime numbers?
Yes, Euclid proved this over 2000 years ago. His elegant proof shows that if you multiply all known primes and add 1, either this new number is prime, or it has a prime factor not in your original list - thus primes cannot be finite.
How are prime numbers used in cryptography?
Public-key cryptosystems like RSA use large primes (typically 1024-4096 bits) because multiplying two large primes is easy, but factoring the product back into primes is computationally infeasible with current technology, securing encrypted communications.