Check if a number is prime and find its factors.
Check whether a number is prime (only divisible by 1 and itself) and find all its factors. Perfect for mathematics education, number theory, cryptography learning, and algorithm testing. Provides prime status, all factors, and factorization details.
Check if a number is prime and find its factors.
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Prime numbers are natural numbers greater than 1 that have exactly two factors: 1 and themselves. For example, 7 is prime because it is only divisible by 1 and 7, while 8 is not prime (composite) because it has factors 1, 2, 4, and 8. Prime numbers are the building blocks of number theory—every integer greater than 1 can be expressed as a unique product of primes (Fundamental Theorem of Arithmetic). For instance, 12 = 2 × 2 × 3, where 2 and 3 are primes.
Testing primality involves checking if a number has any divisors other than 1 and itself. The naive approach tests divisibility by all integers from 2 to n-1, but this is inefficient for large numbers. Optimized algorithms test only up to the square root of n because any factor larger than √n must pair with a factor smaller than √n. For example, testing 49 requires checking divisors only up to 7 (√49 = 7), finding that 49 = 7 × 7. Further optimizations test only prime divisors or use probabilistic algorithms (Miller-Rabin) for very large numbers.
The distribution of primes is irregular yet predictable statistically. Small numbers have relatively many primes (4 in the first 10: 2, 3, 5, 7), but they become sparser as numbers grow. The Prime Number Theorem states that the number of primes less than n is approximately n / ln(n). Despite their scarcity at large scales, there are infinitely many primes (proven by Euclid over 2,000 years ago). Special prime forms include twin primes (pairs differing by 2, like 11 and 13) and Mersenne primes (2^p - 1, where p is prime).
Cryptography relies on prime numbers for secure encryption algorithms like RSA. RSA uses the fact that multiplying two large primes is easy (e.g., 7919 × 7927 = 62,794,313), but factoring the product back to the original primes is computationally hard for very large numbers. Secure RSA keys use primes hundreds of digits long, making factorization infeasible with current technology. Prime checking validates that candidate numbers for cryptographic keys are indeed prime, ensuring encryption strength and security.
Mathematics education uses prime checking to teach number theory, divisibility, and algorithm efficiency. Students learn to identify primes, understand factorization, and explore patterns in prime distribution. Exercises like finding all primes below 100 (the Sieve of Eratosthenes) develop algorithmic thinking. Understanding primes is foundational for advanced topics like modular arithmetic, Fermat's Little Theorem, and public-key cryptography. Prime checkers provide instant feedback, helping students verify manual calculations and explore large-number behavior.
Algorithm optimization and computer science education benefit from prime checking as a classic performance testing problem. Implementing efficient primality tests teaches Big-O notation, algorithmic complexity, and optimization techniques. Students compare naive O(n) algorithms against O(√n) trial division, O(log n) probabilistic tests (Miller-Rabin), and deterministic polynomial algorithms (AKS). Profiling prime checking code reveals how algorithmic improvements drastically reduce computation time, a lesson applicable to all software optimization.
Performance degradation with large numbers makes naive primality tests impractical. Testing if 1,000,000,007 is prime by checking all divisors up to 1 billion takes billions of operations. Optimized algorithms test only up to √n (31,623 for this case), reducing operations by over 30,000×. For very large numbers (hundreds of digits), deterministic tests are too slow, requiring probabilistic methods (Miller-Rabin) that provide near-certainty with minimal computation. Always use sqrt(n) optimization or better for production prime checking.
Edge cases for small numbers (0, 1, 2) require special handling. By definition, primes must be greater than 1, so 0 and 1 are not prime. The number 2 is the only even prime (all other even numbers are divisible by 2), making it a special case. Beginners often forget to handle 2 separately, leading to incorrect results when algorithms skip even numbers. Always implement explicit logic for n < 2 (not prime), n = 2 (prime), and even n > 2 (not prime) before testing odd divisors.
Composite numbers with large smallest factors challenge efficient primality tests. If a composite number's smallest factor is large (e.g., 101 × 103 = 10,403), trial division must test many divisors before finding one. For cryptographic applications where primes must be guaranteed (not probabilistic), deterministic tests are required but slow. Modern solutions use hybrid approaches: fast probabilistic tests (Miller-Rabin) for initial screening, then deterministic verification (Lucas test) for candidates that pass. This balances speed and certainty.
Use sqrt(n) optimization for trial division to minimize unnecessary iterations. Test divisibility only up to the square root of the number because any factor larger than sqrt(n) must pair with a smaller factor already tested. For example, checking if 121 is prime requires testing only divisors up to 11 (sqrt(121)), finding that 11 × 11 = 121. Implement this with a loop: for i from 2 to sqrt(n), check if n % i == 0. This simple optimization makes trial division practical for numbers into the millions.
Skip even divisors after checking for divisibility by 2 to halve iteration count. After verifying that n is not divisible by 2, test only odd divisors (3, 5, 7, 9, ...). This reduces iterations by 50% with minimal code complexity. For further optimization, test divisibility by 2 and 3, then check divisors of the form 6k±1 (5, 7, 11, 13, 17, 19, ...), which encompasses all primes > 3 and skips multiples of 2 and 3. This reduces iterations by 67% compared to naive approaches.
Use probabilistic primality tests (Miller-Rabin) for very large numbers where deterministic methods are too slow. Miller-Rabin provides probabilistic certainty: with k rounds of testing, the probability of incorrectly identifying a composite as prime is less than 4^-k. For k=10 rounds, error probability is less than 1 in a million. Cryptographic applications use k=40+ for near-absolute certainty. For production use, combine fast probabilistic screening with deterministic confirmation for critical numbers, balancing speed and accuracy.
