Prime Number Checker
Instantly verify if any number up to 1,000,000,000,000,000,000 (10¹⁸) is prime or composite. Get exact prime factorization, nearest primes, and deep mathematical insights.
Instant Test Presets:
1,000,000,000,000,000,000
1000000000000000000 can be divided cleanly by factors other than 1 and itself.
Prime Factorization
Distinct Prime Factors:
Nearest Prime Neighbors
Mathematical Properties & Attributes
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 rounding errors.
- BigInt Precision: Safely compute on 64-bit integers up to $10^18$ without floating-point approximations.
- Deterministic Miller-Rabin: Utilizes 12 mathematically proven prime bases for 0% false positives.
- Instant Factorization: Automatically decomposes composite numbers into exponents of prime factors using Pollard's rho algorithm.
- 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 Division
Small Prime Screening
Checks divisibility against small primes (2, 3, 5, 7, ..., 97) to eliminate even and simple composite numbers instantly.
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
Pollard's Rho Algorithm
For composite numbers, extracts non-trivial divisors using polynomial cycle detection until all prime factors are found.
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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 divisors are 1 and itself (e.g., 2, 3, 5, 7, 11, 13, 17, 19). Numbers greater than 1 that are not prime are called composite numbers.
By mathematical convention and the Fundamental Theorem of Arithmetic, 1 is classified as neither prime nor composite. If 1 were considered prime, prime factorization would not be unique (for example, 6 could be factored as 2 × 3, or 1 × 2 × 3, or 1² × 2 × 3).
Standard JavaScript numbers lose precision above 9 quadrillion (2^53 - 1). Our tool uses native JavaScript BigInt arithmetic combined with the deterministic Miller-Rabin primality test. By testing against 12 specific prime bases, it is mathematically guaranteed to give 100% accurate results for all 64-bit integers up to 10^18 in less than a millisecond.
Prime factorization is the process of breaking down a composite number into the product of prime numbers that multiply together to equal the original number. For example, the prime factorization of 1000 is 2 × 2 × 2 × 5 × 5 × 5, written compactly as 2³ × 5³.
No, absolutely not. All primality tests, factorizations, and range calculations execute entirely locally within your web browser using your device's processor. No numbers or data are collected, stored, or transmitted over the internet.