Random Prime Number Generator
Generate random prime numbers (divisible only by 1 and themselves) with customizable ranges - essential for cryptography, number theory, algorithm testing, and mathematical research
Our prime number generator creates random integers divisible only by 1 and themselves within any range from 2 to 1,000,000. Generate prime numbers for cryptographic applications, RSA key generation, hash functions, primality testing algorithms, number theory research, and educational demonstrations. Perfect for mathematical testing, security protocol development, algorithm analysis, and statistical sampling of prime distributions. Prime numbers are fundamental to modern cryptography and include values like 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31... with infinite extent but decreasing density. Use our generator for random prime integers in encryption systems, mathematical proofs, computational challenges, and applications requiring numbers with no divisors except 1 and themselves.
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What is a Prime Number Generator?
A prime number generator produces random integers divisible only by 1 and themselves, with no other factors. Prime numbers are fundamental to number theory and cryptography, defined as any integer p > 1 where the only positive divisors are 1 and p. Our generator creates cryptographically random prime numbers within your specified range from 2 (the smallest prime) to 1,000,000, using efficient primality testing algorithms to ensure mathematical correctness. The sequence begins 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37... and extends infinitely, though primes become less frequent in larger ranges following the Prime Number Theorem: approximately n/ln(n) primes exist below n.
The generator employs optimized primality testing (combining trial division for small candidates and Miller-Rabin probabilistic tests for larger values) to verify each generated number satisfies the prime constraint. Support for unique (non-repeating) generation enables statistical sampling without replacement from the prime distribution, while sorted output produces ascending sequences for mathematical analysis. Use prime generation for RSA cryptography (selecting large primes for public/private key pairs), hash function design (prime moduli improve distribution), algorithm testing (validating prime-specific optimizations), number theory research (studying prime gaps, twin primes, Mersenne primes), educational demonstrations of the Sieve of Eratosthenes, and applications requiring numbers guaranteed to have no divisors beyond trivial factors.
Prime Number Generator Configuration
Range (Min/Max)
Count (1-100 Primes)
Unique (No Duplicates)
Sorted Output
How to Generate Prime Numbers
[STEP 1] Set Range Boundaries
Configure minimum (at least 2) and maximum values to define the prime search space. Consider that prime density decreases with larger ranges. For cryptographic applications, use larger ranges (e.g., 100000-1000000). For educational demonstrations, smaller ranges (e.g., 2-100) are more manageable.
[STEP 2] Configure Count & Options
Choose how many primes to generate (1-100). Enable unique mode for distinct primes without repetition, essential for RSA key pair generation where p and q must differ. Enable sorted mode to analyze prime distribution, gaps, and clustering patterns.
[STEP 3] Generate & Verify Primality
Click EXECUTE GENERATION to create random primes using cryptographically secure algorithms with primality verification. Results display individual primes with metadata showing range, sum, average, and count. Each value is mathematically guaranteed divisible only by 1 and itself.
[STEP 4] Export & Apply
Copy individual primes or export as TXT (one per line), CSV (with index), or JSON (with full metadata). Use in cryptographic systems (RSA, Diffie-Hellman), hash function design, primality testing benchmarks, number theory research, or educational curriculum demonstrating fundamental mathematical properties.
Prime Number Generation Best Practices
- _ Cryptographic Applications - For RSA key generation, use large distinct primes (enable unique mode). RSA security depends on the difficulty of factoring the product of two large primes. Consider ranges 100,000+ for stronger keys.
- _ Range Selection - Prime density decreases logarithmically: ~25% of numbers under 100 are prime, ~7% under 1,000, ~4.3% under 10,000. Choose ranges appropriate for your application to balance prime availability with computational cost.
- _ Unique Mode for Key Pairs - When generating RSA key pairs, ALWAYS enable unique mode to ensure p ≠ q. Identical primes compromise security as n = p² becomes trivially factorable.
- _ Prime Gaps Analysis - Enable sorted mode to study gaps between consecutive primes. Twin primes differ by 2 (e.g., 11 and 13), cousin primes by 4, sexy primes by 6. Prime gaps generally increase with magnitude.
- _ Primality Testing Performance - Our generator uses optimized algorithms: trial division for small numbers, Miller-Rabin for larger values. Generation time increases with range size due to lower prime density requiring more candidates.
- _ Special Primes - Consider specific prime types for specialized applications: Mersenne primes (2ᵖ - 1) for perfect numbers, safe primes (p where (p-1)/2 is also prime) for cryptographic groups, Sophie Germain primes for discrete logarithm security.
Technical Implementation
Our prime number generator employs efficient primality testing algorithms combining trial division for small candidates with Miller-Rabin probabilistic testing for larger values, ensuring cryptographically random selection from the prime distribution:
// Prime Number Generation Algorithm
Algorithm: Random Prime Selection with Primality Verification
// Step 1: Validate range (min >= 2)
if (min < 2) then min = 2
assert min >= 2 (smallest prime)
// Step 2: Build set of all primes in range
primes = empty_set
for candidate = min to max:
if is_prime(candidate) then primes.add(candidate)
// Step 3: Primality testing function
function is_prime(n):
if n <= 1 then return false
if n <= 3 then return true // 2 and 3 are prime
if n mod 2 = 0 or n mod 3 = 0 then return false
// Trial division by numbers of form 6k±1 up to √n
for i = 5 to √n step 6:
if n mod i = 0 or n mod (i+2) = 0 then return false
return true
// Step 4: Random sampling from prime set
if (unique) then shuffle(primes) and take first count
else randomly select count primes (with replacement)
// Step 5: Optional sorting
if (sorted) then sort(results) ascending
// Mathematical Properties:
Definition: p is prime ⟺ p > 1 and divisors(p) = {1, p}
Prime Number Theorem: π(n) ≈ n / ln(n) primes below n
First Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37...