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Prime Number Calculator

4 months ago
Last Update on December 7, 2025
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Prime Number Calculator

🔢 Prime Number Calculator

Advanced prime number checker to test primality, find prime factors, generate prime lists, and analyze number properties. Supports large numbers with multiple algorithms.

✓ Prime Number Checker
Is Prime?
✓ Yes
Prime Status
Number Type
Prime
Classification
Divisors Count
2
Total Divisors

📋 Number Properties

Divisors: 1, 17

Square Root: 4.12

Binary Form: 10001

🔍 Prime Factorization Tool
Prime Factors
2, 3, 5
Unique Factors
Factorization
2² × 3 × 5
Exponential Form

📋 Factorization Details

Factors will appear here...
📊 Prime Generator
Prime Count
0
Total Primes

📋 Generated Primes

Primes will appear here...

💡 About Prime Numbers

A prime number is a natural number greater than 1 that has exactly two divisors: 1 and itself. Prime numbers are fundamental in mathematics and cryptography. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Every integer greater than 1 can be uniquely expressed as a product of prime numbers (Fundamental Theorem of Arithmetic).

Understanding Prime Numbers & Primality Testing

Prime numbers are the building blocks of mathematics and play crucial roles in cryptography, computer science, and number theory. Understanding how to identify primes, factorize composites, and generate prime lists is essential for students, mathematicians, and programmers. This comprehensive calculator provides multiple tools for analyzing and working with prime numbers efficiently.

Prime Number Algorithms & Formulas

Trial Division Primality Test
$$\text{A number } n \text{ is prime if } n > 1 \text{ and has no divisors except 1 and itself}$$
$$\text{Test divisibility for all } d \text{ where } 2 \leq d \leq \sqrt{n}$$
$$\text{Example: For } n = 17, \text{ test } 2 \text{ to } 4: \text{ None divide 17} \Rightarrow \text{Prime}$$
Efficiency: O(√n) time complexity. Practical for numbers up to 10¹⁶.
Sieve of Eratosthenes Algorithm
$$\text{1. List all integers from 2 to } n$$
$$\text{2. Mark multiples of 2 (except 2 itself)}$$
$$\text{3. For each unmarked number } p, \text{ mark its multiples}$$
$$\text{4. Remaining unmarked numbers are prime}$$
Efficiency: O(n log log n). Fastest for generating all primes up to n.
Prime Factorization Formula
$$n = p_1^{a_1} \times p_2^{a_2} \times p_3^{a_3} \times \ldots \times p_k^{a_k}$$
$$\text{Where } p_i \text{ are prime numbers and } a_i \text{ are positive integers}$$
$$\text{Example: } 60 = 2^2 \times 3 \times 5$$
Fundamental Theorem of Arithmetic: Every integer > 1 has unique prime factorization.

Prime Number Distribution & Properties

RangeFirst PrimeLast PrimeCountDensity
1-1027440%
1-1002972525%
1-1,000299716816.8%
1-10,00029,9731,22912.3%
1-100,000299,9919,5929.6%
1-1,000,0002999,98378,4987.8%

Prime Number Examples & Applications

Example 1: Checking if 29 is Prime
Number: 29

Method: Trial Division
• Check divisors from 2 to √29 ≈ 5.39
• Test: 2, 3, 4, 5
• 29 ÷ 2 = 14.5 (not divisible)
• 29 ÷ 3 = 9.67 (not divisible)
• 29 ÷ 5 = 5.8 (not divisible)

Result: 29 is PRIME (divisors: 1, 29)
Example 2: Prime Factorization of 120
Number: 120

Factorization Process:
• 120 ÷ 2 = 60
• 60 ÷ 2 = 30
• 30 ÷ 2 = 15
• 15 ÷ 3 = 5
• 5 ÷ 5 = 1

Result: 120 = 2³ × 3 × 5

Frequently Asked Questions About Primes

What is the smallest prime number?
The smallest prime number is 2. It is the only even prime number, and all other prime numbers are odd. The sequence of first few primes is: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31...
Is 1 considered a prime number?
No, 1 is not considered prime by modern mathematical definition. A prime number must have exactly two distinct positive divisors: 1 and itself. The number 1 only has one divisor (itself), so it doesn't meet the criteria. By convention, the prime numbers begin with 2.
What is the difference between prime and composite numbers?
Prime numbers have exactly 2 divisors (1 and the number itself), while composite numbers have more than 2 divisors. For example, 7 is prime (divisors: 1, 7), but 8 is composite (divisors: 1, 2, 4, 8). Every integer greater than 1 is either prime or composite.
How are primes used in cryptography?
Large prime numbers are fundamental to modern encryption (RSA). The security relies on the fact that while it's easy to multiply two large primes, it's computationally difficult to factor their product back into the original primes. This one-way property makes primes essential for secure communication.
Are there infinitely many prime numbers?
Yes, there are infinitely many prime numbers. This was proven by Euclid around 300 BC. His proof shows that no finite list of primes can be complete—you can always find another prime larger than any given list. Despite their infinite count, primes become increasingly sparse as numbers grow larger.
What is the Fundamental Theorem of Arithmetic?
The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be uniquely represented as a product of prime numbers (up to the order of factors). For example, 60 = 2² × 3 × 5. This is unique—no other prime factorization exists for 60. This theorem is foundational to number theory.

Conclusion

Prime numbers are fascinating mathematical objects with applications ranging from cryptography to fundamental scientific research. Whether you're checking primality, factorizing composites, or generating prime lists, this advanced calculator provides the tools needed for efficient computation. Understanding prime numbers and their properties is essential for mathematics, computer science, and cybersecurity. Use this calculator to explore the fascinating world of primes and develop deeper mathematical insights.

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