๐ Prime Checker
Enter an integer and instantly find out whether it's a prime number. If it's not prime, its prime factorization is shown too. Useful for math homework or learning the basics behind cryptography.
How to use
- Enter the integer you want to check (0 to 1 trillion).
- Whether it's prime, and its prime factorization if not, are shown automatically.
How the calculation works
A prime is an integer of at least 2 whose only divisors are 1 and itself. To test whether n is prime, it is enough to check for divisors from 2 up to โn: if n had a divisor larger than โn, its partner divisor would have to be smaller than โn, so the larger range never needs checking. This tool goes further. After ruling out 2 and 3, it only tries numbers of the form 6k โ 1 and 6k + 1 (5, 7, 11, 13, 17, 19, โฆ). Every prime above 3 has that form, because 6k, 6k+2, 6k+3 and 6k+4 are all divisible by 2 or 3, so this cuts the number of trial divisions to about a third. For composite numbers, the prime factorisation is shown as well, found by trial division โ dividing out each factor as many times as it goes, smallest first.
Worked example
Is 221 prime? 1. โ221 โ 14.87, so checking divisors up to 14 is enough 2. Not divisible by 2 or 3 3. Try the 6kยฑ1 candidates: 5 no, 7 no, 11 no, 13 โ 221 รท 13 = 17, divides exactly 221 is not prime: 221 = 13 ร 17. 223, by the same process, has no divisor up to 14 and is therefore prime.
Things to be aware of
- 1 is not prime; the definition requires an integer of at least 2.
- Integers up to one trillion are supported. Larger values would put too heavy a load on the browser.
- Negative numbers and decimals cannot be tested for primality.
- Testing the hundreds-of-digits primes used in cryptography calls for different methods, such as probabilistic primality tests. This tool is for learning and quick checks.
FAQ
Is 1 a prime number?
No. By definition, a prime number is a natural number greater than 1 with no divisors other than 1 and itself, so 1 doesn't qualify.
How large a number can this check?
Integers up to 1 trillion (1,000,000,000,000) are supported. Larger numbers are excluded since the calculation would become too slow in a browser.
What happens if I enter a negative number or a decimal?
Primality is only defined for natural numbers 2 and above, so no result is shown for negative numbers or decimals.