# Perfect Number Checker

σ(n) = 2n ⟺ perfect. Factorization-based σ(n), proper-divisor sum, deficient/perfect/abundant classification, abundancy index, Euclid–Euler form. n ≤ 10¹².

> Canonical page: https://elysiatools.com/en/tools/perfect-number-checker

- **Category:** Math & Numbers

- **Keywords:** perfect number, aliquot sum, abundant number, deficient number, abundancy index, divisors, mersenne prime, number theory

## Overview

The Perfect Number Checker analyzes integers up to 10¹² to determine whether they are perfect, deficient, or abundant based on prime factorization and proper divisor sums. It computes the sum-of-divisors function σ(n), the abundancy index σ(n)/n, identifies Euclid–Euler form parameters for even perfect numbers, and can list proper divisors.

## Inputs

- **Number n** (text): 1 ≤ n ≤ 10¹² (trial-division factorization bound).
- **Output detail** (select)

## When to use

- Verifying whether a given integer equals the sum of its proper divisors.
- Classifying integers up to 10¹² as deficient, perfect, or abundant alongside their abundancy index.
- Deconstructing known even perfect numbers into their Euclid–Euler form 2^(p−1)·(2^p − 1) and associated Mersenne primes.

## How it works

- Enter an integer n between 1 and 10¹² and select whether to output classification statistics only or include the list of proper divisors.
- The tool performs trial-division prime factorization to determine prime factors and their exponents.
- Using multiplicative properties, it calculates σ(n), derives the proper-divisor sum (σ(n) − n), and determines the abundancy index.
- For perfect numbers, it matches the value against the Euclid–Euler form to display prime exponent p and corresponding Mersenne prime (2^p − 1).

## Use cases

- Checking number theory homework problems involving aliquot sums and divisor functions.
- Exploring properties of small abundant numbers and calculating exact abundancy indices.
- Studying Mersenne primes and validating even perfect numbers within the range up to 10¹².

## Frequently asked questions

### What is a perfect number?

A perfect number is a positive integer that equals the sum of its proper divisors, meaning σ(n) = 2n.

### What is the difference between deficient, perfect, and abundant numbers?

A number is deficient if the sum of its proper divisors is less than n, perfect if equal to n, and abundant if greater than n.

### What is the maximum value of n supported?

The tool supports integers n from 1 up to 10¹² (1,000,000,000,000).

### When does the tool list individual proper divisors?

Proper divisors are listed when the 'Output detail' option is set to include divisors and the total proper divisor count is 200 or fewer.

### How does the tool identify the Euclid–Euler form?

For even perfect numbers, it expresses n as 2^(p−1)·(2^p − 1), verifying the exponent p and Mersenne prime factor.

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