Prime Number Checker
Instantly verify if any integer up to 10³² (33 digits / 100 nonillion) is prime or composite. Get exact prime factorization, nearest primes, twin prime generation, and deep mathematical insights.
1,000,000,000,000,000,000
1000000000000000000 can be divided cleanly by factors other than 1 and itself.
Prime Factorization
Distinct Factors:
Nearest Prime Neighbors
Mathematical Attributes & Insights
Even (2n)
19 Digits
1
Yes (1000000000²)
Yes (1000000³)
No
No
0
Advanced Primality & Factorization Engine
Whether you are solving number theory homework, verifying cryptographic primes, or exploring mathematics, our Prime Number Checker delivers exact answers instantly without server latency or precision loss.
- BigInt Precision: Safely compute on large integers up to $10^32$ (up to 33 digits) without floating-point approximations.
- Deterministic Miller-Rabin: Utilizes 12 mathematically proven witness bases for 0% false positives up to $1.84 \times 10^19$ and ultra-fast probabilistic testing for numbers up to $10^32$.
- Fast Factorization: Decomposes composite numbers using Brent's cycle-finding variant of Pollard's rho algorithm and instant root detection.
- 100% Private & Offline: Runs entirely in your browser; turn off your internet and keep calculating securely.
Why Prime Numbers Matter
Primes are the building blocks of modern mathematics and computing:
- Cryptography & Security: RSA and Diffie-Hellman encryption rely on the computational difficulty of factoring massive prime numbers.
- Fundamental Theorem: Every integer greater than 1 has a unique prime factorization, acting as its mathematical fingerprint.
- Hash Tables & RNGs: Prime numbers are frequently used in computer algorithms to minimize hash collisions and generate pseudo-random sequences.
- Number Theory: From Mersenne primes to Twin Primes and the Riemann Hypothesis, primes hold the deepest mysteries in science.
How Our Algorithm Works
When you enter a number, our tool performs a multi-stage analysis:
Stage 1 · Fast Screening
Small Prime Division
Checks divisibility against small primes (2, 3, 5, 7, ..., 97) to eliminate even and simple composite numbers in microseconds.
Stage 2 · Miller-Rabin
Deterministic Primality
Applies modular exponentiation against 12 exact witness bases to guarantee 100% determinism up to $1.84 \times 10^19$.
Stage 3 · Factorization
Brent's Pollard Rho
For composite numbers, extracts non-trivial divisors using Brent's polynomial cycle detection and instant root checks with bounded execution.
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Frequently Asked Questions
A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. In other words, its only positive divisors are 1 and itself (e.g., 2, 3, 5, 7, 11, 13, 17, 19). Numbers greater than 1 that have more than two divisors are called composite numbers.
By mathematical definition and the Fundamental Theorem of Arithmetic, 1 is classified as a unit (neither prime nor composite). If 1 were prime, prime factorization would not be unique (e.g., 6 could be factored as 2 × 3, or 1 × 2 × 3, or 1² × 2 × 3). Zero has infinitely many divisors (since 0 × k = 0 for all integers k) and is neither prime nor composite.
Standard JavaScript numbers lose precision above 9 quadrillion (2^53 - 1). Our tool uses native JavaScript BigInt arithmetic combined with the optimized Miller-Rabin primality test. By testing against 12 mathematically proven witness bases {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37}, it evaluates integers up to 10³² (33 digits / 100 nonillion) with astronomical precision in less than a millisecond.
For composite numbers, our engine first filters out small prime factors up to 97, checks for instant perfect square and cube powers (O(1)), and then extracts non-trivial divisors using Brent's cycle-finding variant of Pollard's rho algorithm with batch GCD. All calculations are strictly bounded to ensure instantaneous UI responsiveness.
Twin Primes are pairs of prime numbers that differ by exactly 2 (such as 3 and 5, 5 and 7, 11 and 13, 17 and 19). The Generator tab lets you discover all prime numbers or filter strictly for Twin Primes within any custom range up to a 5,000 number span at once.
No, absolutely not. All primality tests, factorizations, nearest prime searches, and range generation execute entirely locally within your web browser using client-side JavaScript. No numbers or data are ever collected, stored, or transmitted over the internet.