Prime Number Checker
Test primality instantly using Miller-Rabin algorithm. Get prime factorization, divisors, twin primes, and complete number theory analysis up to 10¹⁵.
Free Prime Number Checker: Test Primality & Factorization Online Instantly
Check if numbers are prime using advanced Miller-Rabin algorithm. Get instant prime factorization, divisors, twin primes, Sophie Germain primes, Mersenne primes, and complete number theory analysis up to 10¹⁵. Perfect for students, mathematicians, cryptography enthusiasts, and educators.
What Is a Prime Number (And Why Prime Testing Matters)?
A prime number is a natural number greater than 1 that has exactly two divisors: 1 and itself. Prime numbers are the fundamental building blocks of number theory—every integer can be uniquely expressed as a product of primes (Fundamental Theorem of Arithmetic). Understanding primality is crucial for cryptography, computer security, hash functions, and mathematical research according to Wikipedia's Prime Number article.
Professional prime number testing uses sophisticated algorithms like Miller-Rabin (probabilistic primality test) and Trial Division (deterministic for small numbers). Our tool implements both methods with smart algorithm selection—using fast Trial Division for numbers under 1,000 and deterministic Miller-Rabin with 12 witnesses for larger numbers up to 10¹⁵, providing 100% accuracy with millisecond performance.
Why Prime Number Testing Is Essential for Modern Technology:
Cryptography & Security
- • RSA encryption: Uses large primes (2048+ bits) for secure communication
- • Key generation: Finding safe primes protects against attacks
- • Digital signatures: Prime-based algorithms verify authenticity
- • Blockchain: Cryptographic hashing relies on prime properties
Education & Research
- • Number theory: Study prime gaps, distributions, patterns
- • Mathematics education: Teach factorization and divisibility
- • Algorithm analysis: Benchmark primality testing methods
- • Computer science: Understand computational complexity
Real Prime Number Examples
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 First 15 prime numbers—only divisible by 1 and themselves4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25 Have more than two divisors (not prime)Mersenne: 2³¹ - 1 = 2,147,483,647 Form 2ᵖ - 1 where p is prime—used in cryptography982,451,653 (9-digit prime) Verified using Miller-Rabin algorithm in <1msHow to Check Prime Numbers in 3 Simple Steps
💡 Pro Tip: Finding Safe Primes for Cryptography
A safe prime is a prime number p where (p-1)/2 is also prime (called Sophie Germain prime). Check if a number is Sophie Germain prime using our detector—these primes are critical for Diffie-Hellman key exchange and digital signature algorithms. For example, a small Sophie Germain prime when doubled and incremented by one yields another prime. Use our tool to verify cryptographic primes before implementing encryption systems.
7 Number Theory Features Our Prime Checker Performs
Uses 12 witness values to deterministically verify primality for all numbers up to 3.3×10²¹ (covers entire int64 range). Based on Miller-Rabin algorithm with optimized modular exponentiation. Time complexity O(k log³ n) where k=12 witnesses. Catches all composites with 100% accuracy while running in milliseconds for 15-digit numbers.
Tests divisibility by all numbers up to √n for guaranteed primality. Optimized with early exit on even numbers and checking only odd divisors. For numbers under 1,000, this method is faster than Miller-Rabin due to lower computational overhead. Our tool automatically selects Trial Division for n < 1,000, ensuring optimal performance across all input sizes.
Decomposes composite numbers into prime factors with multiplicity. Uses optimized trial division: handle 2 separately, then test odd numbers up to √n, dynamically updating the square root as factors are removed. Displays results in mathematical notation (e.g., 360 = 2³ × 3² × 5) with Unicode superscripts. Essential for understanding number structure and the Fundamental Theorem of Arithmetic.
Calculates all divisors of a number using efficient pair finding: for each i from 1 to √n, if n mod i = 0, both i and n/i are divisors. Returns sorted divisor list, count (τ function), and sum (σ function). For large numbers (>1,000,000), uses formula-based calculation from prime factorization to avoid memory constraints. Crucial for perfect number detection and number theory research.
Twin Primes: Detects pairs (p, p+2) or (p-2, p) where both are prime (e.g., 11 and 13, 17 and 19). Sophie Germain Primes: Identifies primes p where 2p+1 is also prime—critical for cryptographic security. Mersenne Primes: Verifies form 2ᵖ - 1 where exponent p is prime. These rare primes are used in random number generation and the largest known primes are always Mersenne.
Classifies numbers based on sum of proper divisors (all divisors except n itself). Perfect: sum equals n (e.g., 6 = 1+2+3, 28 = 1+2+4+7+14)—extremely rare, only 51 known. Abundant: sum > n (e.g., 12 < 1+2+3+4+6=16). Deficient: sum < n (all primes are deficient). Uses efficient σ(n) calculation from prime factorization for large numbers instead of brute-force summation.
Finds previous and next prime numbers relative to input using optimized search (test odd candidates backward/forward until prime found). Calculates prime index π(n) for small primes—the position of a prime in the sequence. This helps identify which prime in the ordered list corresponds to a given value. Shows prime gaps (distance between consecutive primes) which average ~ln(n) but can be arbitrarily large. Essential for studying Prime Number Theorem and prime distributions.
10 Real-World Prime Number Applications
1. RSA Cryptography & Public Key Infrastructure
Generate large prime numbers (2048-4096 bits) for RSA key pairs used in SSL/TLS certificates, SSH authentication, and PGP encryption. Test smaller primes to understand the algorithm before implementing production systems. Our tool helps verify Sophie Germain primes for enhanced security. Combine with our SSL certificate checker for complete security analysis.
2. Mathematics Education & Student Learning
Teach number theory concepts like prime factorization, GCD/LCM calculation, divisibility rules, and the Fundamental Theorem of Arithmetic. Students can explore prime patterns, test conjectures (Goldbach's conjecture: every even number greater than two is sum of two primes), and understand computational complexity. Perfect for K-12 mathematics, discrete math courses, and competitive programming preparation. Use with our scientific calculator for complete math toolkit.
3. Algorithm Performance Benchmarking
Test and compare primality testing algorithms (Trial Division vs Miller-Rabin vs AKS) for different input sizes. Our tool displays computation time in milliseconds, allowing performance analysis. Benchmark your own prime checking implementations against our optimized code. Study time complexity practically by testing increasing number sizes and observing algorithmic behavior.
4. Competitive Programming & Coding Interviews
Verify solutions to prime-related problems on LeetCode, HackerRank, Codeforces, and interview questions. Common problems: count primes up to n (Sieve of Eratosthenes), find nth prime, check primality in O(√n), factorize numbers, find prime gaps. Use our batch checker to validate test cases quickly and understand expected outputs for edge cases like unity, the first prime, large primes, and Mersenne numbers.
5. Hash Function Design & Computer Science Research
Use prime numbers in hash table sizing (table size should be prime to minimize collisions), polynomial rolling hash functions, and universal hashing. Verify prime moduli for cryptographic hash functions and checksums. Research prime-based algorithms for randomization, probabilistic data structures (Bloom filters use multiple hash functions with prime parameters), and string matching algorithms. Check our hash generator tool for cryptographic hashing.
6. Number Theory Research & Mathematical Discovery
Explore unsolved problems: Riemann Hypothesis (related to prime distribution), twin prime conjecture (infinitely many twin primes?), Goldbach's conjecture verification, prime gaps analysis. Test Mersenne prime candidates (form 2ᵖ - 1, test if p and result are prime). Find Sophie Germain prime chains. Analyze prime density in ranges. Study arithmetic progressions of primes with constant difference.
7. Blockchain & Cryptocurrency Security
Verify prime numbers used in elliptic curve cryptography (secp256k1 for Bitcoin uses prime 2²⁵⁶ - 2³² - 977). Test primality of curve parameters and field sizes. Understand modular arithmetic over prime fields. Study proof-of-work algorithms that may involve prime-based calculations. Essential for blockchain developers and crypto security auditing.
8. Random Number Generation & Monte Carlo Simulation
Prime numbers are used in Linear Congruential Generators (LCG) for pseudo-random number generation: Xₙ₊₁ = (aXₙ + c) mod m, where m should be prime. Verify prime moduli for better randomness properties. Test Mersenne Twister parameters (uses Mersenne primes). Critical for simulations, gaming, statistics, and cryptographic random number generators requiring high-quality randomness.
9. Distributed Systems & Load Balancing
Use prime numbers in consistent hashing algorithms to distribute data across servers. Prime-based hash functions minimize clustering when assigning keys to nodes. Verify prime ring sizes for distributed hash tables (DHT). Test modulo prime operations for round-robin load balancing with better distribution properties. Reduces hotspots in CDNs, databases, and cache systems using our developer tools.
10. Coding Challenge Solutions & Project Euler
Solve Project Euler problems involving primes: sum of primes below n, largest prime factor, circular primes, truncatable primes, prime permutations, prime digit replacements. Verify solutions for Advent of Code, Google Code Jam, and ICPC problems. Test performance of Sieve of Eratosthenes implementations. Generate prime sequences for algorithm testing and validation.
7 Prime Number Testing Mistakes to Avoid
1. Testing Even Numbers Without Quick Check
All even numbers except 2 are composite—always check if n = 2 first, then if n is even (n mod 2 = 0), immediately return false. This optimization eliminates 50% of inputs instantly. Many beginners test divisibility by all numbers without this early exit, wasting computational resources on guaranteed composites.
2. Checking Divisors Up to n Instead of √n
Trial division only needs to check up to √n because divisors come in pairs: if n = a × b and a ≤ b, then a ≤ √n. Checking all numbers up to n is O(n) instead of O(√n)—for n = 1,000,000, that's 1 million checks vs 1,000 checks, making the algorithm 1000× slower. Always use sqrt limit for efficiency.
3. Not Handling Edge Cases (Zero, Unity, Negative Numbers)
By definition, prime numbers are natural numbers greater than unity. Many implementations fail on edge cases: zero and unity are neither prime nor composite, negative numbers are undefined for primality. Always validate input: if n < 2, return false. Missing edge cases causes incorrect results and fails test cases in competitive programming.
4. Using Probabilistic Tests Without Enough Witnesses
Basic Miller-Rabin with 1-2 witnesses has false positives (Carmichael numbers pass some witness tests). Using k = 12 specific witnesses makes the test deterministic for all 64-bit integers. Random witnesses give probabilistic guarantee with error probability 4⁻ᵏ, but deterministic witness sets eliminate all errors. Our tool uses 12 proven witnesses for 100% accuracy.
5. Integer Overflow in Modular Arithmetic
Miller-Rabin requires computing (a × b) mod m for large numbers. Direct multiplication a × b can overflow even 64-bit integers. Use careful modular multiplication: break into smaller operations or use specialized overflow-safe algorithms. Our implementation handles multiplication safely for all int64 values without overflow errors that produce wrong primality results.
6. Inefficient Factorization for Large Numbers
Brute-force factorization testing all numbers up to n is impossibly slow for large composites. Use optimized trial division: test 2 separately, then only odd numbers up to √n, updating the square root as factors are found. for semi-primes (product of two large primes), even optimized trial division takes too long—use Pollard's rho or quadratic sieve for cryptographic-sized numbers.
7. Not Caching Prime Lists for Repeated Checks
When checking many numbers (batch processing, sieve implementations), regenerating the same small primes is wasteful. Use Sieve of Eratosthenes to pre-compute primes up to √(max_input), then test divisibility only by these primes. for checking primes up to 10⁶, generate all primes ≤ 1,000 once and reuse—reduces 1,000 divisibility tests to ~168 (number of primes ≤ 1,000).
Frequently Asked Questions About Prime Numbers
What's the largest prime number ever discovered?
As of 2024, the largest known prime is the Mersenne prime 2⁸²,⁵⁸⁹,⁹³³ - 1, discovered in December 2018 by the Great Internet Mersenne Prime Search (GIMPS). It has over 24 million digits—printing it would require thousands of pages. Our tool can verify Mersenne prime candidates up to the maximum 64-bit integer value. All record-breaking primes since 1996 have been Mersenne primes.
How accurate is Miller-Rabin primality testing?
Basic Miller-Rabin with random witnesses is probabilistic with error probability ≤ 4⁻ᵏ for k witnesses. However, using specific deterministic witness sets makes it 100% accurate for certain ranges. Our implementation uses 12 proven witnesses that guarantee correctness for all 64-bit integers (up to 2⁶³ - 1 ≈ 9.2×10¹⁸). For numbers in this range, our tool never gives false positives or false negatives—it's completely deterministic and reliable.
Why are there no even prime numbers except 2?
A prime number has exactly two divisors: unity and itself. All even numbers are divisible by two, so they have at least three divisors (unity, two, and the number itself)—making them composite by definition. The number two is special: it's the only even prime because its only divisors are unity and itself. This makes two the "oddest" prime—it's the only even one and the smallest prime number.
What are Sophie Germain primes and why do they matter?
A Sophie Germain prime is a prime p where 2p + 1 is also prime (called a safe prime). Examples include small primes like 2, 3, and 5 where doubling and adding one yields another prime. They're crucial for cryptography—safe primes are used in Diffie-Hellman key exchange to prevent small-subgroup attacks. The discrete logarithm problem is harder over groups with safe prime order, making encryption stronger. Sophie Germain primes are relatively rare: only about 3,000 are known with more than 5,000 digits.
How do I find the next prime number after a given number?
Start from n + 1 and test each candidate for primality until a prime is found. Optimization: if n is even, start from n + 1 (odd); if n is odd, start from n + 2 and only test odd numbers. Our tool finds the next prime by testing candidates using Miller-Rabin, typically finding it within a few tests based on the Prime Number Theorem. The average gap between consecutive primes grows logarithmically with the size of the numbers.
What is a Mersenne prime and how do I check for one?
A Mersenne number has the form Mₚ = 2ᵖ - 1 for prime p. If Mₚ is also prime, it's a Mersenne prime. Small examples include values where the exponent is a small prime number. To verify: first check the exponent p is prime, then check if 2ᵖ - 1 is also prime. Our tool verifies both conditions. Only 51 Mersenne primes are known; they're the largest primes discovered because Lucas-Lehmer test makes them easier to verify than random primes.
Can I use this tool for RSA key generation?
Our tool handles numbers up to 10¹⁵ (50 bits), which is insufficient for secure RSA (requires 2048+ bit primes). However, it's excellent for learning RSA concepts: test small primes p and q, compute n = p×q, verify φ(n) = (p-1)(q-1), choose public exponent e coprime to φ(n). for production RSA, use specialized libraries like OpenSSL or cryptography.io that generate cryptographically secure primes. Our tool helps you understand the mathematics before implementing real systems.
How often should I use batch prime checking vs single checks?
Use single mode when you need detailed analysis: full factorization, divisor lists, special prime detection, and number classification. Use batch mode for quickly checking if multiple numbers are prime/composite without detailed analysis—perfect for filtering lists, testing sequences, or validating algorithm outputs. Batch mode processes up to 100 numbers simultaneously with results in tabular format for easy comparison.
Advanced Prime Number Techniques
Sieve of Eratosthenes for Batch Generation
Generate all primes up to n efficiently: create boolean array, mark multiples of each prime as composite. Time: O(n log log n). Perfect for Project Euler, competitive programming, and generating prime lists. Much faster than testing each number individually for ranges—find all primes under 1,000,000 in milliseconds.
Wheel Factorization Optimization
Skip multiples of small primes entirely. Instead of testing every odd number, test only numbers coprime to small composites. This wheel factorization eliminates most composite candidates, making trial division significantly faster for large ranges while adding minimal overhead.
Probabilistic Compositeness Testing
For cryptographic applications requiring billions of primality tests, use Fermat test first (a^(n-1) ≡ 1 mod n for witness a) to quickly eliminate most composites, then run full Miller-Rabin on survivors. Reduces computation by ~90% for large-scale prime searching.
Prime Gap Analysis for Distribution Study
Study gaps between consecutive primes: g(n) = pₙ₊₁ - pₙ. Average gap ~ ln(n), but maximum gaps grow. Record gaps have been documented in the hundreds and thousands. Use our tool to find gaps in sequences, verify Cramér's conjecture, and explore prime clustering patterns for research.
Pollard's Rho for Large Factorization
When factoring large semi-primes (products of two primes), trial division is too slow. Pollard's rho algorithm finds factors in O(n¹/⁴) expected time using cycle detection. Essential for breaking weak RSA keys or factoring numbers with ~20+ digits where trial division becomes impractical.
Carmichael Number Detection
Carmichael numbers are composite but pass Fermat primality test for all bases coprime to n. Known examples include several three and four-digit numbers. They're pseudoprimes that fool basic tests. Miller-Rabin with multiple witnesses catches all Carmichaels—our 12-witness implementation detects them with 100% accuracy, preventing false positives in cryptographic applications.
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Test primality instantly with Miller-Rabin algorithm. Get complete factorization, divisor analysis, special prime detection, and number theory insights. Check numbers up to 10¹⁵—100% free, no signup required, privacy-focused, educational tool.
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