Prime Number Test Calculator

This prime number test checks whether a whole number is prime, meaning it has no divisors other than 1 and itself. Enter a number up to a billion and the calculator tells you whether it is prime or composite, and if it is composite it shows the smallest number that divides it. A prime number is one of the basic objects of number theory, important in cryptography, hashing and many algorithms. The test uses trial division by 2 and then the odd numbers up to the square root, which is enough to settle primality exactly. The method, a worked example and the assumptions are set out below.

Whole number (up to 1,000,000,000)

Smallest divisor (if composite)

Classification

The formula

A number n greater than 1 is prime if no integer from 2 up to the square root of n divides it evenly. If any does, n is composite and that divisor is a proper factor.

Worked example

97 is not divisible by 2, 3, 5 or 7, and the square root of 97 is under 10, so there is nothing left to test: 97 is prime. By contrast 91 = 7 times 13, so its smallest divisor is 7. Enter 97 or 91 to confirm.

Frequently asked questions

What makes a number prime?

It is a whole number above 1 whose only divisors are 1 and itself, such as 2, 3, 5, 7 and 11.

Is 1 prime?

No. By definition a prime has exactly two distinct divisors. The number 1 has only one, so it is neither prime nor composite.

How does the test work?

It tries dividing by 2 and the odd numbers up to the square root of n. If none divide evenly, n is prime.

Who this calculator is for

This calculator is for students, programmers and anyone exploring number theory.

What this calculator assumes

You enter a whole number within the stated range.
The exact integer algorithm is used.
Very large results are shown grouped for readability.

Formula and sources

Related calculators

Prime Factorization Calculator: break a number into prime factors.
LCM Calculator: lowest common multiple.
GCF Calculator: greatest common factor.

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