Test whether an integer is prime with an exact deterministic check across the 64-bit range. Inspect its prime factorization when practical, divisor statistics, digit properties, and the nearest prime on either side.
Step by step
01Enter an integer
Signs and grouping separators are accepted; primality applies to positive integers greater than one.
02Read the verdict
Prime means exactly two positive divisors. Composite means the number can be written as a product of smaller positive integers.
03Inspect the factorization
For manageable values, see each prime power and the resulting divisor count and sum.
04Look at the neighbors
Use the previous and next primes to understand the local gap or pick a nearby prime value.
Worked example
Given
A divisor below the square root is enough
221 ÷ 13 = 17
221 = 13 × 17
divisors: 1, 13, 17, 221A composite number must have a factor no larger than its square root. Finding 13 settles the question without testing every number below 221.
Why this one
One is neither prime nor composite
A prime has exactly two positive divisors, while one has only itself. Keeping one separate preserves unique prime factorization.
Trial division is not the only test
Miller–Rabin uses modular exponentiation to reject composites quickly; a fixed witness set makes it deterministic for 64-bit inputs.
A primality answer can be easier than factoring
Tests can certify that a large integer is composite without cheaply revealing all of its factors.
Negative signs do not create negative primes
Standard elementary number theory defines primes among positive integers greater than one; factorization can still discuss the magnitude.
The judgement call
Need only a yes or no on a large 64-bit value
Primality test
Modular witnesses answer without enumerating factors.
Simplifying a fraction
GCD
Shared divisibility matters more than full factorization.
Counting all divisors
Prime factors
Exponents give the divisor count directly.
Cryptographic key generation
Specialized audited library
A browser utility is educational, not a secure key-generation workflow.
Reference
FAQ
No. One has only one positive divisor, while a prime must have exactly two: 1 and itself.
Under the standard definition, no. Primes are positive integers greater than one, though the magnitude of a negative integer has the same positive prime factors.
The prime number checker uses deterministic Miller–Rabin with a witness set proven sufficient for 64-bit integers, using exact modular arithmetic.
Primality testing and finding every factor are different computational tasks. The former can be fast even when trial division for the latter would take too long.
It is the smallest prime greater than that number. The tool checks successive eligible integers with the same exact primality test.
Primality tests and factor calculations run locally. The integer is not uploaded.
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