Prime Number & Factorization

Decomposing composite integers into irreducible prime components forms the bedrock of modern public-key cryptography, hashing algorithms, modular arithmetic, and discrete mathematics. The user supplies a single positive integer to evaluate. The tool performs primality testing and, for composite numbers, systematically breaks the value down into a unique product of prime bases raised to exponential powers in accordance with the Fundamental Theorem of Arithmetic.

The output clearly identifies whether the entered value is prime or composite and displays its full canonical factorization in exponent notation. This breakdown allows students and programmers to identify all fundamental divisors, simplify square root radicals, or construct common denominators for rational arithmetic. A key mathematical caveat to remember is that the number one is defined as a unit rather than a prime or composite number, and negative numbers must be stripped of their sign prior to integer factorization.

Primality is tested by performing trial division against potential prime factors up to the square root of the target integer. When factors are found, the algorithm divides the number iteratively and records the multiplicity of each prime base. The final output expresses the number in canonical form as the product of prime factors raised to their respective powers.