A prime number has exactly two factors: 1 and itself. This checker tests if your number can be divided by any number other than 1 and itself. Numbers less than 2 are not considered prime.
The prime number checker determines whether a number is prime (has exactly two factors: 1 and itself) or composite (has more than two factors). Choose your mode: Quick Check (fast yes/no answer) or Show Factors (displays all factors for composite numbers).
Enter any positive integer in the input field. Try small numbers first: 7 is prime, 8 is composite. The checker works instantly for numbers up to millions. Type 17 and click Check to verify it's prime.
For Quick Check mode, you get a simple result: "17 is a PRIME number" in green, or "18 is NOT a prime number (composite)" in orange. This mode is fastest when you just need a yes/no answer.
Switch to Show Factors mode to see why a number is composite. Try 12: the checker shows it's composite and lists all factors: 1,2,3,4,6,12. This helps understand the number's structure. Use Reset to test another number.
Using Quick Check Mode
Select Quick Check for the fastest primality test. This mode applies efficient algorithms to determine if your number has exactly two factors. The result is binary: prime or composite.
The checker tests divisibility up to the square root of your number. For 49, it only needs to test up to 7 because 49=7. Finding that 7 divides 49 immediately confirms it's composite.
Try testing known primes: 2,3,5,7,11,13,17,19,23,29. The result shows green "PRIME" for each. Now try 4,6,8,9,10—all show orange "NOT prime (composite)" because they have factors beyond 1 and themselves.
Quick Check is perfect for verifying calculations, cryptography work (finding large primes), or number theory problems. It handles large numbers efficiently—try 97 or 101 to confirm they're prime.
Using Show Factors Mode
Switch to Show Factors mode to see all factors when a number is composite. This educational mode helps you understand why a number isn't prime and reveals its factorization structure.
For prime numbers, the factors list shows only 1 and the number itself. Try 13: factors are 1,13—exactly two factors, confirming primality. For composite numbers, you see the complete factor list.
Enter 24 in Show Factors mode: the checker displays "24 is NOT a prime number (composite)" and lists factors: 1,2,3,4,6,8,12,24. Eight factors total means it's highly composite (very factorable).
The factor count tells you about the number's properties. Perfect squares like 16 have an odd number of factors (1,2,4,8,16 = five factors) because the square root pairs with itself. Try 36: nine factors.
Understanding Prime Numbers
A prime number is a positive integer greater than 1 that has exactly two factors: 1 and itself. The number 7 is prime because only 1×7=7—no other factor pairs exist. You cannot build 7 from smaller whole numbers.
The first prime numbers are: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47.... Notice 2 is the only even prime—all other even numbers are divisible by 2, giving them at least three factors.
Composite numbers have more than two factors. The number 12 is composite with factors 1,2,3,4,6,12. Every composite number can be broken into prime factors: 12=2×2×3=22×3.
The number 1 is special—it's neither prime nor composite. By definition, primes need exactly two factors, but 1 has only one factor (itself). Zero and negative numbers are not considered in standard prime number theory.
How the Prime Test Works
The checker uses trial division: testing if any number from 2 to n divides n evenly. If a divisor is found, n is composite. If none exists, n is prime.
Why only test up to n? If n=a×b and a>n, then b<n. So checking up to n catches all factor pairs. For 100: test up to 10, finding 2,4,5 divide evenly—composite.
The algorithm starts with 2, then tests odd numbers 3,5,7,9... up to n. Even numbers greater than 2 are skipped after checking 2 because they're automatically composite. This optimization speeds up the test.
For very large numbers, more sophisticated tests exist (Miller-Rabin, AKS primality test), but trial division works perfectly for numbers up to millions that this calculator handles. The efficiency is O(n) complexity.
Reading Results and Factor Lists
Results appear color-coded for clarity. Green text "PRIME" indicates exactly two factors. Orange text "NOT prime (composite)" indicates three or more factors. Red error messages show invalid input.
The factors list (in Show Factors mode) is always ordered from smallest to largest: 1,2,3... up to the number itself. The count shows how many factors exist. For 12: "6 factors found (composite number)".
Prime numbers always show exactly two factors: 1 and the number. Try 31: factors shown are 1,31. The factor count for primes is always 2. Any other count proves the number is composite.
Perfect squares have a special factor pattern. The middle factor equals n and pairs with itself. For 25: factors are 1,5,25. The factor 5 appears once because 5×5=25. This creates an odd factor count.
Prime Numbers in Mathematics
Prime numbers are the building blocks of all integers. Every number greater than 1 is either prime or can be uniquely factored into primes. This is the Fundamental Theorem of Arithmetic: 60=22×3×5.
Twin primes are prime pairs differing by 2: (3,5),(5,7),(11,13),(17,19),(29,31). An unsolved problem asks if infinitely many twin primes exist. The largest known twin primes have hundreds of thousands of digits.
Mersenne primes have the form 2p−1 where p is prime. Examples: 3=22−1, 7=23−1, 31=25−1. These are used to find record-breaking large primes. The largest known prime (282,589,933−1) is a Mersenne prime with 24+ million digits.
Primes have no simple formula—they appear irregularly among integers. The Prime Number Theorem describes their density: roughly ln(n)n primes exist below n. But predicting specific primes remains difficult.
Applications of Prime Numbers
Cryptography: RSA encryption uses very large prime numbers (hundreds of digits). The security relies on difficulty of factoring products of two huge primes. Your online banking uses this prime-based encryption.
Hash tables: Prime-sized tables reduce collisions in computer science data structures. Hash functions work better with prime moduli because primes minimize common factors with data patterns.
Random number generation: Linear congruential generators use prime numbers as moduli to create pseudorandom sequences. The randomness quality depends on choosing appropriate primes.
Cicada emergence: Some cicada species emerge every 13 or 17 years—both prime numbers. This evolutionary strategy minimizes overlap with predators operating on different cycles, avoiding common multiples.
Music and harmony: The frequency ratios in musical harmony often involve small primes (2:3 for perfect fifth, 3:4 for perfect fourth). Prime relationships create pleasing consonance because they minimize interference patterns.
Special Cases and Edge Numbers
The number 1: Not prime (only one factor) and not composite (definition requires at least two prime factors). Historically sometimes called prime, but modern mathematics excludes it for cleaner theorems.
The number 2: The only even prime and the smallest prime. All other even numbers are divisible by 2, making them composite. This makes 2 unique in prime theory.
The number 0: Not considered in prime number theory. Zero is divisible by every number (since n×0=0 for all n), which doesn't fit the prime definition.
Negative numbers: Standard prime theory uses only positive integers. While mathematically you could extend concepts to negatives, the conventional definition restricts primes to numbers greater than 1.
Large numbers: As numbers grow, primes become rarer (though infinite in quantity). Between 1−100: 25 primes. Between 1,000,000−1,000,100: only 6 primes. The gaps between consecutive primes grow larger.
Related Calculators and Concepts
Factoring Calculator - Find all factors or prime factors of any number. Shows the complete prime factorization that proves a number is composite. Essential for understanding number structure.
GCF Calculator - Greatest common factor of two+ numbers. Prime numbers have GCF = 1 with any number not divisible by them. Useful for understanding coprimality.
Divisibility Calculator - Test if numbers divide evenly. Prime numbers are only divisible by 1 and themselves. Divisibility testing is the core of primality checking.
LCM Calculator - Least common multiple. For two primes p and q, LCM(p,q)=p×q because they share no factors. Shows the unique relationship between coprime numbers.
Modulo Calculator - Find remainders. Related to prime testing through Fermat's Little Theorem: if p is prime and a not divisible by p, then ap−1≡1(modp).