About

Number theory relies on understanding the fundamental properties of integers. Whether for cryptography (RSA encryption keys), algorithm optimization, or academic research, knowing if a number is prime, perfect, or highly composite is essential. This tool performs a deep analysis of any input integer. It utilizes trial division optimized with a pre-computed sieve for speed, followed by robust factorization algorithms for larger values. It provides instantaneous conversions between decimal, binary, and hexadecimal systems, offering a complete profile of the number.

Formulas

A positive integer n is analyzed based on the Fundamental Theorem of Arithmetic, which states every integer greater than 1 is either a prime or a product of primes unique up to order:

n = p₁^a₁ × p₂^a₂ × ⋯ × p_k^a_k

A number is considered Perfect if the sum of its proper divisors Sum(d) equals n.

Reference Data

Number (n)	Prime?	Prime Factorization	Binary (Z₂)	Hex (Z₁₆)	Property
6	No	2 × 3	110	6	Perfect
28	No	2² × 7	11100	1C	Perfect
37	Yes	37	100101	25	Prime
100	No	2² × 5²	1100100	64	Square
496	No	2⁴ × 31	111110000	1F0	Perfect
1024	No	2¹⁰	10000000000	400	Power of 2
2025	No	3⁴ × 5²	11111101001	7E9	Square
8128	No	2⁶ × 127	1111111000000	1FC0	Perfect

Frequently Asked Questions

This tool operates within the JavaScript safe integer limit (2^53 - 1, approx 9 quadrillion). Inputting larger numbers may result in precision loss, although the logic will attempt to handle BigInts where compatible for factorization of larger integers.

Many cryptographic protocols (like RSA) rely on the computational difficulty of factoring large composite numbers into their prime components. Understanding how a number breaks down is fundamental to understanding key generation.

A perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For example, 6 has divisors 1, 2, and 3. Since 1 + 2 + 3 = 6, it is a perfect number.

Properties like "Prime" or "Perfect" are generally defined for positive integers (Natural numbers). If you enter a negative number, the tool focuses on its absolute value for factorization logic but notes the sign in the binary representation (Two's complement is not shown here; raw magnitude conversion is used).