Prime Number Calculator

A prime number is a whole number greater than one with exactly two positive divisors: one and the number itself. Numbers greater than one that have additional divisors are composite. Seven is prime because only one and seven divide it evenly, while nine is composite because three is an additional divisor. Zero and one are neither prime nor composite, a convention that preserves important number-theory results and keeps prime factorization unique. This calculator tests primality by looking for a divisor. Even numbers greater than two can be rejected immediately. For an odd candidate, only odd divisors up to its square root need to be checked. If a composite number equals a times b, at least one of those factors must be no larger than the square root; otherwise both would be larger and their product would exceed the number. Reaching that boundary without finding a divisor proves the candidate is prime. The result also displays the nearest smaller and larger primes. Finding them means testing consecutive integers outward until a prime is reached. Two has no smaller prime because it is the first prime number. Prime gaps vary: some primes are only two apart and form twin-prime pairs, while other neighboring primes have much larger differences. There is always another prime above any chosen integer, as Euclid's classic proof shows that the sequence of primes never ends. Prime numbers are central to arithmetic because every integer greater than one factors uniquely into primes. They support fraction simplification, greatest common divisor methods, modular arithmetic, and many divisibility arguments. Modern public-key cryptography uses operations built around large primes and computationally difficult inverse problems. Hash tables, randomization techniques, error-correcting codes, and cyclic patterns also draw on prime properties. For hand checking, first eliminate divisibility by two, three, and five, then continue with possible prime divisors no greater than the square root. Divisibility rules can make those checks faster, but trial division remains the underlying proof for modest inputs. The calculator applies exact integer remainder tests rather than a statistical estimate, so every answer within the accepted range is deterministic. JavaScript represents ordinary numbers exactly only through its safe integer limit, so the input is restricted to that range. Extremely large cryptographic candidates require specialized big-integer algorithms such as Miller-Rabin followed by stronger proof methods. For schoolwork, factor checks, sequence exploration, and everyday number theory, the exact square-root method used here is straightforward and reliable. Enter any nonnegative integer to classify it and see its nearest prime neighbors.