Prime Number Checker
Check if any integer is prime instantly — with step-by-step trial division proof for small numbers and deterministic Miller-Rabin for large numbers. Shows the previous prime, next prime, prime gaps, and the next 5 primes after your number. Supports integers up to 10³⁰ — free, private, and no signup required.
Enter any integer to instantly check if it is prime. Shows trial division steps for small numbers, Miller-Rabin for large numbers, and displays the nearest primes. All calculations run locally in your browser — no signup required.
Supports integers up to 10³⁰ digits. Commas and underscores are ignored.
Why Use Our Prime Number Checker?
Instant Prime Number Check
Enter any integer up to 10³⁰ and instantly find out if it is prime. The prime number checker uses trial division for small numbers and the deterministic Miller-Rabin test for large numbers — giving you a mathematically certain answer in milliseconds.
Secure Prime Number Checker Online
The prime number checker runs entirely client-side in your browser. Your input values are never sent to any server, stored, or tracked — complete privacy for cryptography, academic, and research use.
Prime Number Checker — No Installation
Use the prime number checker directly in any modern browser with no downloads, apps, or plugins required. Load famous primes and composites with one click, check the next 5 primes, and navigate between primes — 100% free forever.
Step-by-Step Proof & Nearest Primes
The prime number checker shows the full trial division steps for small numbers — every divisor tested with its remainder — so you can follow the proof. It also shows the previous prime, next prime, prime gap, and the next 5 primes after your number.
Common Use Cases for Prime Number Checker
Cryptography & Security
Cryptographers need large prime numbers for RSA key generation, Diffie-Hellman key exchange, and elliptic curve cryptography. Use the prime number checker to verify that candidate primes are truly prime before using them in cryptographic protocols.
Mathematics Education
Students learning number theory use the prime number checker to explore prime distribution, verify homework answers, and understand the trial division algorithm. The step-by-step proof makes it an ideal teaching tool for primality testing.
Programming & Algorithm Development
Developers implementing prime sieves, hash functions, and number theory algorithms use the prime number checker to validate their implementations. Check edge cases like 1, 2, large Mersenne primes, and Carmichael numbers instantly.
Competitive Programming
Competitive programmers verify prime numbers during contest problem solving. The prime number checker handles numbers up to 10³⁰ — far beyond what most contest problems require — and shows the nearest primes for prime gap problems.
Number Theory Research
Mathematicians studying prime gaps, twin primes, and prime distributions use the prime number checker to quickly verify primality and find neighboring primes. The next-5-primes feature makes it easy to explore prime sequences.
Puzzle & Recreational Mathematics
Math enthusiasts checking whether interesting numbers (like 1729, the Hardy-Ramanujan number, or 561, the first Carmichael number) are prime use the prime number checker for instant verification with a clear explanation.
Understanding Prime Number Checking
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 few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29... The number 1 is not prime by definition. The number 2 is the only even prime — all other even numbers are divisible by 2 and therefore composite. A composite number is any integer greater than 1 that is not prime — it has at least one factor other than 1 and itself. Our prime number checker determines whether any integer is prime or composite and shows the mathematical proof, using trial division for small numbers and the Miller-Rabin algorithm for large numbers.
How Our Prime Number Checker Works
- Enter Your Number: Type any integer into the prime number checker. Numbers up to 10³⁰ are supported. Commas and underscores are ignored, so you can enter 2,147,483,647 or 2_147_483_647. Click any preset button to load a famous prime or composite number instantly.
- Primality Test:Click “Check Prime” and the prime number checker selects the best algorithm automatically. For numbers up to 10¹⁵, it uses trial division — testing divisors of the form 6k±1 up to √n — and shows every step. For larger numbers, it uses the deterministic Miller-Rabin test with 12 witnesses, which is proven correct for all n < 3.3 × 10²⁴.
- View Results: The prime number checker displays the verdict (PRIME or NOT PRIME), the smallest factor if composite, the previous and next primes with their prime gaps, the next 5 primes after your number, and the full step-by-step proof — all processed locally in your browser.
Trial Division vs. Miller-Rabin
- Trial Division: Tests whether n is divisible by any integer from 2 to √n. Only needs to check divisors of the form 6k±1 (after handling 2 and 3 separately), reducing the work by a factor of 3. Exact and shows every step, but slow for very large numbers.
- Miller-Rabin Primality Test:A probabilistic test that, with carefully chosen witnesses, becomes deterministic. With the 12 witnesses used by our prime number checker (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37), the test is deterministic for all n < 3,317,044,064,679,887,385,961,981.
- Carmichael Numbers: Numbers like 561 = 3 × 11 × 17 pass some primality tests but are composite. The Miller-Rabin test with multiple witnesses correctly identifies all Carmichael numbers as composite.
- Mersenne Primes: Primes of the form 2ⁿ − 1, such as 2³¹ − 1 = 2,147,483,647. These are among the largest known primes. The prime number checker correctly identifies them using Miller-Rabin.
Prime Gaps
A prime gap is the difference between two consecutive prime numbers. The gap after 2 is 1 (to 3), after 3 is 2 (to 5), and gaps generally grow as numbers get larger — though irregularly. The prime number checker shows the gap to both the previous and next prime. The largest known prime gaps are studied in number theory as part of understanding the distribution of primes. By the Prime Number Theorem, the average gap near n is approximately ln(n).
Related Tools
Voice Recorder & Audio Extractor
Record high-quality audio from your microphone or extract audio from MP4 and WebM video files offline. 100% secure, browser-based utility.
Audio Slicer & Converter
Trim audio clips and convert between MP3, WAV format client-side - Free online audio cutter
ID3 Tag & Metadata Editor
Read and write ID3 tags, album art, artist, and track details directly to MP3 file headers - Free online ID3 tag editor
MP3 Metadata Viewer
View all ID3 tags — title, artist, album, artwork, BPM, and every embedded frame — from any MP3 file instantly in your browser - Free online MP3 metadata viewer
Frequently Asked Questions About Prime Number Checker
A prime number checker determines whether a given integer is prime (divisible only by 1 and itself) or composite (has other factors). Our prime number checker supports integers up to 10³⁰, shows step-by-step trial division proof, displays the nearest primes, and runs entirely in your browser — no signup required.
No. By mathematical convention, 1 is neither prime nor composite. The definition of a prime number requires it to be greater than 1 and have exactly two distinct positive divisors: 1 and itself. The number 1 has only one positive divisor (itself), so it does not qualify.
Yes. 2 is prime — it is divisible only by 1 and 2. It is also the only even prime number, since all other even numbers are divisible by 2 and therefore composite. The prime number checker correctly identifies 2 as prime.
For numbers up to 10¹⁵, the prime number checker uses trial division — testing divisors of the form 6k±1 up to √n — and shows every step. For larger numbers, it uses the deterministic Miller-Rabin primality test with 12 witnesses (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37), which is proven correct for all n < 3.3 × 10²⁴.
A Carmichael number is a composite number that passes the Fermat primality test for all bases coprime to it — making it a "pseudoprime." The smallest Carmichael number is 561 = 3 × 11 × 17. The Miller-Rabin test with multiple witnesses correctly identifies all Carmichael numbers as composite.
A Mersenne prime is a prime of the form 2ⁿ − 1. For example, 2³¹ − 1 = 2,147,483,647 is a Mersenne prime. Not all numbers of this form are prime — 2⁴ − 1 = 15 = 3 × 5 is composite. The prime number checker correctly identifies Mersenne primes using Miller-Rabin.
Yes. The prime number checker runs 100% locally in your browser. Your input values are never sent to any server, stored in a database, or tracked in any way. Everything stays completely private on your device.
Yes — the prime number checker is 100% free with no signup, no account, and no usage limits. Check prime numbers as many times as you need, completely free forever. There are no ads, no premium tiers, and no data collection.
The prime number checker supports integers up to 10³⁰ (a 30-digit number). For numbers above 10¹⁵, it uses the Miller-Rabin test which is deterministic for all n < 3.3 × 10²⁴. For numbers between 10²⁴ and 10³⁰, the test is probabilistic with an extremely low error probability.