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Home › Math & Conversions › Core Math & Algebra › Prime number Prime Number Calculator Check whether a number is prime, factor composite integers, and list primes in a range – with clear explanations and number theory context. Core Math & Algebra Interactive prime number tool Use this calculator to test if a number is prime, see its prime factorization if composite, or generate a list of primes within a given interval. Designed for students, teachers, and anyone working with number theory or cryptography basics. Prime check & factorization Prime list in range Integer to test Enter a positive integer ≥ 2. For very large values, computation may take a bit longer. Quick tips Even numbers > 2 are not prime. Numbers ending in 5 and > 5 are not prime. You only need to test divisors up to \(\sqrt{n}\). Check primality Clear Load example: large composite Result summary – Classification Prime factorization Divisibility reasoning Start of range End of range Guidelines For performance and readability, keep the range length ≤ 100 000. The tool will truncate the displayed list after 1 000 primes while still reporting the full count. List primes in range Clear Primes in range Summary Prime list What is a prime number? A prime number is a natural number greater than 1 that has exactly two positive divisors: 1 and itself. Numbers greater than 1 that are not prime are called composite , because they can be written as a product of smaller positive integers. The first few prime numbers are: \[ 2,\;3,\;5,\;7,\;11,\;13,\;17,\;19,\;23,\;29,\dots \] Note that 2 is the only even prime. Every other even number is divisible by 2 and is therefore composite. Prime factorization and the building blocks of integers A key fact from elementary number theory is the Fundamental Theorem of Arithmetic : Every integer \(n > 1\) is either prime or can be written as a product of prime numbers in a way that is unique up to the order of the factors. For example: \(60 = 2^2 \cdot 3 \cdot 5\) \(84 = 2^2 \cdot 3 \cdot 7\) \(131\) is prime, so its prime factorization is just \(131\) itself. The factorization returned by the calculator is written in a standard prime-power format, such as \(2^3 \cdot 7\), which is convenient for later computations with greatest common divisors, least common multiples, and modular arithmetic. How to test if a number is prime For a given integer \(n \ge 2\), a simple primality test is: If \(n = 2\) or \(n = 3\), it is prime. If \(n\) is divisible by 2 or 3 and greater than 3, it is composite. Otherwise, test divisibility by all integers of the form \(6k \pm 1\) up to \(\sqrt{n}\). If none of these integers divides \(n\), then \(n\) is prime. This strategy is efficient enough for educational and everyday use on the range of values supported by this calculator. Prime number density (how often primes occur) Intuitively, prime numbers become less frequent as numbers grow larger, but they never run out. The Prime Number Theorem states that the number of primes up to \(x\), usually denoted \(\pi(x)\), is asymptotically \(\displaystyle \pi(x) \sim \frac{x}{\log x}\) This means that around a large number \(x\), the probability that a random integer is prime is roughly \(\frac{1}{\log x}\). The summary shown by the prime-range tool reports the empirical density of primes in your chosen interval, which you can compare with this heuristic. Prime number – FAQ Why is 1 not considered a prime number? If 1 were prime, it would break the uniqueness part of the Fundamental Theorem of Arithmetic, because you could insert extra factors of 1 into any factorization. By definition, primes have exactly two positive divisors, which rules out 1 (it has only one positive divisor: itself). Are negative numbers prime? In elementary number theory, primes are usually restricted to positive integers greater than 1. For negative integers, one can talk about units and associate each negative integer with its positive counterpart, but the standard definition of a prime number refers to natural numbers only. Does this tool handle extremely large primes like those used in cryptography? No. This calculator focuses on pedagogical ranges: classroom-sized numbers, contest problems, and quick checks. Modern cryptographic applications rely on specialized algorithms such as probabilistic primality tests and large integer arithmetic libraries, which are designed specifically for very large integers. What is a twin prime? A twin prime pair is a pair of primes that differ by 2, such as (3,5), (5,7), (11,13), or (17,19). It is an open problem in mathematics whether there are infinitely many twin primes. How can I use this calculator in teaching or exams? In teaching, the tool is helpful for quickly checking answers and exploring patterns (for example, how primes are distributed in different ranges). In exams, students are usually expected to show their own reasoning and divisibility tests; the calculator is best used as a companion for practice and verification before or after formal assessments. Frequently Asked Questions What are some quick mental checks for non-prime numbers? Several shortcuts help you discard many composite numbers instantly: any even number greater than 2 is not prime; any number ending in 5 and greater than 5 is divisible by 5; and numbers whose digits sum to a multiple of 3 are divisible by 3. Combining these rules can quickly filter out common composites before more detailed tests. Should I always find the full factorization to decide if a number is prime? No. To determine if a number is prime, it is sufficient to show that no prime up to its square root divides it. Full factorization is needed only when you want the exact prime factors of a composite number, which this calculator provides when feasible. How can I use prime factorization in practical problems? Prime factorization is a powerful tool for computing greatest common divisors, least common multiples, simplifying fractions, working with modular arithmetic, and understanding divisibility constraints in combinatorics and probability problems. Related Core Math & Algebra tools Deepen your number theory and algebra skills with these companion tools. Set theory basics Interval notation Base converter Exponent rules Fibonacci number Continued fraction Dice roll probability Pythagorean theorem SOHCAHTOA (trigonometry) Convolution All Math & Conversions tools Prime number checklist Confirm \(n\) is an integer and \(n \ge 2\). Eliminate trivial composites: check divisibility by 2, 3, and 5. Test divisors up to \(\sqrt{n}\) using the \(6k\pm1\) pattern. Record prime factorization if any divisor is found. For large or security-critical values, use specialized primality testing libraries.
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