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Home Page > Math > Basic Math Operations

Perfect Number Checker

MiniWebtool’s Perfect Number Checker compares a positive integer with the sum of its proper divisors. Divisor pairs and a visual breakdown explain whether the number is perfect, abundant, or deficient.

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Perfect Number CheckerTry it now — free ▼
Stack the proper divisors: they match n only for a perfect number

Enter a positive integer below to see whether it is perfect, abundant or deficient, with its proper divisors and their sum.

Try these numbers:

✦ Perfect 6 28 496 8128
✦ Abundant 12 18 945
✦ Deficient 7 15 100

Embed Perfect Number Checker Widget

Video Guide: Perfect Number Checker

Video guide with full transcript

About Perfect Number Checker

Welcome to the Perfect Number Checker, an interactive tool for exploring one of the oldest and most beautiful concepts in number theory. Enter any positive integer to instantly discover whether it is perfect, abundant, or deficient. The tool calculates all proper divisors, displays animated visualizations, and provides a step-by-step mathematical breakdown of the classification.

At a Glance

Perfect Number Checker tests whether a positive integer is perfect, abundant or deficient by comparing it with the sum of its proper divisors. It lists the divisors and shows the working.

Perfect Number Checker: features at a glance
FeatureWhat you getGood for
InputEnter any positive integer, or click a quick example button to load a well-known number.Testing a number from a homework or puzzle
ClassificationThe sum of proper divisors is compared with the number: equal is perfect, greater is abundant, smaller is deficient.Learning how numbers are classified in number theory
Divisor listAll proper divisors are calculated and shown, including divisor pairs.Seeing exactly which divisors make up the sum
ChartsA divisor bar chart and a doughnut comparison show the divisor sum against the number.Visualizing why a number is perfect, abundant or deficient
Worked stepsA step-by-step mathematical breakdown explains the classification.Students checking their own working
Notable numbersA reference table lists examples such as 6, 28, 496 and 8,128 (perfect), 12 (smallest abundant) and 945 (smallest odd abundant).Finding known perfect and abundant numbers

What is a Perfect Number?

A perfect number is a positive integer that is equal to the sum of its proper divisors (all positive divisors excluding the number itself). Perfect numbers are exceptionally rare and have fascinated mathematicians for over 2,000 years, since the time of the ancient Greeks.

Perfect Number Definition
$$n \text{ is perfect if } \sigma(n) = 2n$$

where \(\sigma(n)\) is the sum of all divisors of \(n\)

The first few perfect numbers are:

Abundant and Deficient Numbers

Every positive integer falls into exactly one of three categories based on how the sum of its proper divisors compares to the number itself:

Most numbers are deficient. All prime numbers are deficient (their only proper divisor is 1). The smallest abundant number is 12, whose divisors 1 + 2 + 3 + 4 + 6 = 16 > 12.

The Connection to Mersenne Primes

One of the most remarkable results in number theory, proven by Euler, establishes that every even perfect number has the form:

Euclid-Euler Theorem
$$n = 2^{p-1}(2^p - 1)$$

where \(2^p - 1\) is a Mersenne prime

This means finding new perfect numbers is equivalent to finding new Mersenne primes (primes of the form \(2^p - 1\)). As of 2024, only 52 Mersenne primes are known, corresponding to 52 known even perfect numbers.

Do Odd Perfect Numbers Exist?

Whether any odd perfect number exists is one of the oldest unsolved problems in mathematics. No odd perfect number has ever been found, and it has been proven that if one exists, it must be greater than \(10^{1500}\), have at least 101 prime factors counting repeats, and have at least 10 distinct prime factors. Most mathematicians believe none exist, but a proof remains elusive.

The Abundancy Index

The abundancy index of a number \(n\) is defined as \(\sigma(n)/n\), where \(\sigma(n)\) is the sum of all divisors of \(n\) (including \(n\) itself). This ratio provides a continuous measure of how "abundant" or "deficient" a number is:

How to Use This Calculator

  1. Enter a number: Type any positive integer into the input field, or click a quick example button to try a famous number.
  2. Check the number: Click "Check Number" to compute the proper divisors and their sum.
  3. View the classification: See whether your number is perfect, abundant, or deficient in the animated hero banner.
  4. Explore the visualization: Review the divisor bar chart, doughnut comparison, divisor pairs, and step-by-step calculation.

Notable Numbers

Perfect Number Checker: Notable Numbers
NumberTypeDivisor SumNotable For
6Perfect6Smallest perfect number
12Abundant16Smallest abundant number
28Perfect282nd perfect number
496Perfect4963rd perfect number
945Abundant975Smallest odd abundant number
8,128Perfect8,1284th perfect number

Frequently Asked Questions

What is a perfect number?

A perfect number is a positive integer that equals the sum of its proper divisors (all positive divisors excluding itself). For example, 6 is perfect because 1 + 2 + 3 = 6, and 28 is perfect because 1 + 2 + 4 + 7 + 14 = 28.

What is the difference between abundant and deficient numbers?

An abundant number has proper divisors that sum to more than the number itself (e.g., 12: 1+2+3+4+6=16 > 12). A deficient number has proper divisors that sum to less than the number (e.g., 8: 1+2+4=7 < 8). Perfect numbers are the rare cases where the sum equals the number exactly.

How many perfect numbers are known?

As of 2024, only 52 perfect numbers are known. All known perfect numbers are even. Whether any odd perfect numbers exist remains one of the oldest unsolved problems in mathematics. The largest known perfect number has over 82 million digits.

What is the connection between perfect numbers and Mersenne primes?

Euler proved that every even perfect number has the form \(2^{p-1} \times (2^p - 1)\), where \(2^p - 1\) is a Mersenne prime. Conversely, every Mersenne prime generates a perfect number via this formula. So finding new perfect numbers is equivalent to finding new Mersenne primes.

What is the abundancy index?

The abundancy index of a number \(n\) is the ratio \(\sigma(n)/n\), where \(\sigma(n)\) is the sum of all divisors of \(n\) (including \(n\) itself). Perfect numbers always have an abundancy index of exactly 2. Abundant numbers have an index greater than 2, and deficient numbers have an index less than 2.

Additional Resources

Reference this content, page, or tool as:

"Perfect Number Checker" at https://MiniWebtool.com/perfect-number-checker/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Apr 16, 2026

Developer API available: Run this tool from your app, automation, or agent with one JSON HTTP request. View API docs

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