Prime Number Checker

A prime number is a whole number greater than 1 that has no divisors other than 1 and itself. This calculator checks whether any given number is prime by testing divisibility up to its square root. It also finds the complete prime factorisation of composite numbers.

The calculator can list all primes within a range (e.g. all primes between 1 and 1,000) and find the next prime above or below a given number. Prime factorisation is useful in cryptography, simplifying fractions, and finding the least common multiple (LCM) or greatest common divisor (GCD) of two numbers.

What is a prime number? A whole number greater than 1 with exactly TWO divisors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31. Non-primes (composite): 4 (= 2×2), 6 (= 2×3), 8, 9, 10. Number 1 is NOT prime (definition). Number 2 is the ONLY even prime. All other primes are odd. Prime gap: distance between consecutive primes increases on average but irregularly (twin primes 11/13, then 17/19, then 29/31).

Why are primes important? Building blocks of all integers (Fundamental Theorem of Arithmetic — every integer >1 has unique prime factorisation). Used in: cryptography (RSA encryption uses product of two large primes); error-correcting codes; random number generation; hash functions; computer science. Modern internet security relies on difficulty of factoring large semiprimes (product of two primes). RSA-2048 (current standard) uses 617-digit primes.

Common prime number tests. Trial division: check if N divisible by any number from 2 to √N. Sufficient for small numbers but slow for large. Sieve of Eratosthenes: efficient for finding all primes up to N (eliminate multiples). Miller-Rabin probabilistic test: very fast for large numbers, gives high-confidence primality answer. AKS deterministic test (2002): polynomial-time prime test — theoretically important but slower in practice than Miller-Rabin.

Famous prime number facts. Largest known prime (2024): 2^136,279,841 − 1, a Mersenne prime with 41 million digits. Found via Great Internet Mersenne Prime Search (GIMPS) — distributed computing. Twin primes conjecture: are there infinitely many pairs like (11,13), (17,19), (29,31)? Conjectured YES but unproven. Goldbach's conjecture: every even number >2 is sum of two primes. Verified to 4 × 10^18 but unproven. Riemann hypothesis: about distribution of primes — $1M Millennium Prize unclaimed.

UK A-level Maths prime number topics. Fundamental Theorem of Arithmetic (unique prime factorisation). Highest Common Factor (HCF/GCD) and Lowest Common Multiple (LCM). Euclidean algorithm for GCD. Modular arithmetic and Fermat's Little Theorem (a^p ≡ a mod p for prime p). Public-key cryptography basics (RSA). Number theory in undergraduate Maths: deeper coverage of primes, Diophantine equations, congruences, quadratic residues. Cryptography pathways in UK universities (Bristol, Royal Holloway, Birmingham).