The number theory prime test determines if any integer up to 10,000,000 is prime, finds its prime factorization, lists all divisors, and shows the nearest primes above and below. Supports twin prime detection and prime gap calculation.
Number Theory Prime Test
Enter an integer from 2 to 10,000,000
All Divisors
How to Use the Number Theory Prime Test
Prime numbers are the building blocks of integers. Every positive integer greater than 1 is either prime (divisible only by 1 and itself) or composite (a product of primes). The prime test calculator tells you which category a number falls into.
How Primality Testing Works
For numbers up to 10,000,000, trial division is efficient. Check if any integer from 2 to √n divides n. If none do, n is prime. For n = 97: √97 ≈ 9.85. Check 2,3,4,...,9. None divide 97, so it's prime. For n = 360: 360 = 2³ × 3² × 5.
Prime Factorization
The Fundamental Theorem of Arithmetic states every integer > 1 has a unique prime factorization. The calculator expresses composites in factored form: 360 = 2³ × 3² × 5. The number of divisors = (3+1)(2+1)(1+1) = 24. To find all divisors, take all combinations of the prime powers.
Twin Primes
Twin primes are pairs differing by 2: (3,5), (5,7), (11,13), (17,19), (29,31)... The Twin Prime Conjecture states infinitely many such pairs exist, but it remains unproven. The calculator detects twin prime pairs for your tested number.
FAQ
What is a prime number?
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The first primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29... There are infinitely many primes (proven by Euclid around 300 BC).
How do you test if a number is prime?
For small numbers: trial division — check if any integer from 2 to √n divides n. If none do, n is prime. For large numbers, probabilistic tests like Miller-Rabin are used, which run in milliseconds even for numbers with hundreds of digits.
What is prime factorization?
Prime factorization expresses a composite number as a product of primes. Example: 360 = 2³ × 3² × 5. The Fundamental Theorem of Arithmetic states every integer > 1 has a unique prime factorization (up to ordering).
Is this tool free?
Yes, completely free with no signup required. All calculations run in your browser.
What are twin primes?
Twin primes are pairs of primes that differ by 2: (3,5), (5,7), (11,13), (17,19), (29,31), (41,43)... The Twin Prime Conjecture states there are infinitely many such pairs, but this remains unproven. The largest known twin primes have over 388,000 digits.