CL
Back to Calculators

Prime Number Calculator - Check Primes & Find Prime Factors

Free online prime number calculator. Check if a number is prime, find prime factorization, discover the next prime, or list all primes in a range. Fast and accurate.

8 min read

iAbout This Calculator

Prime numbers are the building blocks of mathematics. A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. Our free prime number calculator helps you quickly determine if a number is prime, break down any number into its prime factors, find the next prime after any number, or generate a list of all primes within a specified range. Whether you're a student learning number theory, a programmer working on cryptography, or just curious about primes, this tool provides instant, accurate results.

?How to Use

  1. 1

    Select the mode from the dropdown: Check if Prime, Prime Factorization, Find Next Prime, or List Primes in Range.

  2. 2

    Enter your number in the input field. For range mode, enter both the start and end numbers.

  3. 3

    Click Calculate to see the result instantly.

  4. 4

    Review the step-by-step breakdown to understand how the calculation was performed.

  5. 5

    Use the results for homework, research, programming, or mathematical exploration.

fFormula

n \text{ is prime if } n > 1 \text{ and } \forall d \in [2, \sqrt{n}], n \mod d \neq 0

A number n is prime if it is greater than 1 and has no divisors between 2 and the square root of n. We only need to check up to the square root because if n has a factor larger than its square root, it must also have a smaller factor.

n
The number being tested for primality
d
Potential divisor being checked
sqrt(n)
The square root of n, the upper bound for checking divisors

Examples

Check if 97 is Prime

Inputs: number: 97, mode: check
Yes, 97 is prime

97 has no divisors between 2 and 9 (sqrt of 97), so it is prime.

Prime Factorization of 120

Inputs: number: 120, mode: factorize
2^3 x 3 x 5

120 = 2 x 2 x 2 x 3 x 5 = 8 x 15 = 120

Find Next Prime After 100

Inputs: number: 100, mode: findNext
101

101 is the first prime number greater than 100.

List Primes Between 10 and 50

Inputs: number: 10, rangeEnd: 50, mode: listRange
11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

There are 11 prime numbers between 10 and 50.

Use Cases

Cryptography and Security

Prime numbers are fundamental to modern encryption algorithms like RSA. Large prime numbers are used to generate secure keys for protecting sensitive data.

Mathematics Education

Students learning number theory, divisibility, and factorization use prime calculators to verify answers and understand prime number properties.

Programming and Algorithms

Developers implement prime-checking algorithms for hash functions, random number generators, and various computational problems.

Scientific Research

Researchers in mathematics and computer science study prime distributions, twin primes, and other number-theoretic phenomena.

Puzzle Solving

Many mathematical puzzles and competitions involve prime numbers, from finding special primes to solving factorization challenges.

Frequently Asked Questions

What is a prime number?
A prime number is a natural number greater than 1 that cannot be divided evenly by any number other than 1 and itself. For example, 7 is prime because its only divisors are 1 and 7.
Why is 1 not a prime number?
By definition, prime numbers must be greater than 1 and have exactly two distinct positive divisors. The number 1 has only one divisor (itself), so it doesn't qualify as prime.
What is prime factorization?
Prime factorization is the process of breaking down a number into its prime factors. For example, 60 = 2 x 2 x 3 x 5. Every positive integer greater than 1 has a unique prime factorization.
How do I find the next prime number?
To find the next prime after a number n, check each subsequent number (n+1, n+2, ...) until you find one that is prime. Our calculator does this automatically and efficiently.
What is the largest known prime number?
The largest known primes are Mersenne primes of the form 2^p - 1. As of 2024, the largest known prime has over 24 million digits. Our calculator works with standard-sized integers.

Conclusion

Prime numbers are fascinating mathematical objects with practical applications in cryptography, computer science, and beyond. Our prime number calculator makes it easy to explore primes, from simple primality tests to complete factorizations. Bookmark this tool for quick access whenever you need to work with prime numbers.