Prime Number Calculator
Check if numbers are prime, find prime factorization, list all primes in a range, or find the nth prime number. Essential tool for number theory and cryptography.
What is a prime number and what makes it special?
A prime number is a natural number greater than 1 that has exactly two factors: 1 and itself. Cannot be divided evenly by any other number. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. NOT prime: 1 (only one factor), 4 (factors: 1,2,4), 6 (factors: 1,2,3,6). Special: 2 is only even prime. Primes are "building blocks" of all numbers via prime factorization. Fundamental to number theory, cryptography (RSA encryption), and mathematics.
How do you check if a number is prime?
Methods: 1) Trial division: Test divisibility by all numbers from 2 to sqrtn. Example: Is 29 prime? sqrt29 ≈ 5.4. Test 2,3,5 - none divide evenly, so 29 is prime. Why sqrtn? If n=a*b and a<=b, then a<=sqrtn. 2) Quick checks: Even numbers (except 2) aren't prime. Sum of digits divisible by 3? Not prime (except 3). Ends in 5 (except 5)? Not prime. 3) For large numbers: probabilistic tests (Miller-Rabin). Optimization: only test prime divisors.
What is prime factorization and how do you find it?
Prime factorization breaks a number into prime number factors. Every number has unique prime factorization (Fundamental Theorem). Method: Divide by smallest prime repeatedly. Example: 60 = 2*30 = 2*2*15 = 2*2*3*5 = 2^2*3*5. Factor tree: 60→6,10→2,3,2,5. Steps: 1) Divide by 2 until odd, 2) Try 3,5,7... up to sqrtn, 3) Remaining number is prime. Uses: GCF/LCM, simplifying fractions, cryptography, finding divisors.
How many prime numbers are there and what patterns exist?
Infinite primes (Euclid's proof ~300 BC). No largest prime! Current largest known: 2⁸^2⁵⁸�^1�^1^3^3⁻^1 (24+ million digits, Mersenne prime). Patterns: No simple formula generates all primes. Gaps increase as numbers grow, but infinitely many twin primes conjectured (like 11,13 or 17,19). Prime density: ~n/ln(n) primes below n. Example: ~168 primes below 1000, ~78,498 below 1,000,000. Distribution appears random but has deep mathematical structure (Riemann Hypothesis).
What are twin primes, Mersenne primes, and other special primes?
Twin primes: Differ by 2 (3,5), (5,7), (11,13), (17,19), (29,31). Infinitely many? Unknown (Twin Prime Conjecture). Mersenne primes: Form 2ᵖ-1 where p is prime. Examples: 3(2^2-1), 7(2^3-1), 31(2⁵-1), 127(2⁷-1). Largest known primes are Mersenne. Sophie Germain primes: p where 2p+1 also prime (2,3,5,11,23). Palindromic primes: Same backwards (11,101,131,151). Emirp: Prime whose reverse is different prime (13↔31, 17↔71).
Why is 1 not considered a prime number?
By definition, primes must have exactly TWO distinct factors. 1 has only ONE factor (itself). Historical: 1 was sometimes called prime, but modern math excludes it. Reason: Fundamental Theorem requires UNIQUE prime factorization. If 1 were prime, 12 = 2^2*3 = 1*2^2*3 = 1^2*2^2*3 (infinite factorizations). Also breaks many theorems and formulas. Convention settled ~20th century. Remember: 2 is smallest prime (and only even one).
How are prime numbers used in cryptography and security?
RSA encryption (internet security) relies on: 1) Easy to multiply large primes (p*q), 2) Extremely hard to factor product back. Example: p=61, q=53, n=3233 is public. Finding p,q from 3233 is hard for huge primes. Real RSA uses 300+ digit primes. Product has 600+ digits. Best factoring methods take centuries for large numbers. Also used in: Diffie-Hellman key exchange, digital signatures, blockchain, random number generation. Quantum computers threaten this (Shor's algorithm).
What is the Sieve of Eratosthenes and how does it work?
Ancient algorithm (240 BC) to find all primes up to n. Steps: 1) List numbers 2 to n, 2) Mark 2 as prime, cross out multiples (4,6,8...), 3) Next unmarked is 3, cross multiples (6,9,12...), 4) Repeat with 5,7... up to sqrtn, 5) Remaining are prime. Example for n=30: Keep 2,3,5,7,11,13,17,19,23,29. Efficient for finding many primes at once. Used in calculator's list function. Modern variants: segmented sieve, Sieve of Atkin.
What real-world applications use prime numbers?
Beyond cryptography: 1) Cicadas emerge in 13 or 17 year cycles (prime) to avoid predator synchronization. 2) Hash tables - prime-sized tables reduce collisions. 3) Music - some rhythms use prime patterns. 4) Graphics - random number generators. 5) Error detection codes (CRC checksums). 6) Clock arithmetic in computers. 7) Distribution algorithms. 8) GPS signal processing. 9) Quantum physics - energy levels. 10) Financial models - random walk simulations. Primes appear throughout nature and technology.
How do you find the nth prime number efficiently?
No direct formula, must generate/check primes sequentially. Methods: 1) Sieve of Eratosthenes to upper bound, extract nth. Upper bound estimate: p_n ≈ n*ln(n) for n>5. Example: 100th prime ≈ 100*ln(100) ≈ 461 (actual: 541). 2) Generate and test primes until count reaches n. 3) For large n: Use prime number theorem refinements. Known values: 10th=29, 100th=541, 1000th=7919, 10000th=104729. Small primes memorized helps. Calculator uses trial division up to reasonable limits.